





f 












* 



PRINCIPLES OF MECHANISM. 

A TREATISE ON 

THE MODIFICATION OF MOTION 

BY MEANS OP THE 

ELEMENTARY COMBINATIONS OF MECHANISM, 
OR OF THE PARTS OF MACHINES. 



FOB USE IN COLLEGE GLASSES, 

BY MECHANICAL ENGINEEBS, 

ETC., ETC. 



BY 

STILLMAN W. ROBINSON, C.E., D.Sa, 

Mechanical Engineer and Expert for the Wire Grip Fastening Co. ; Vice-President 
and Mechanical Engineer to the Grip Machinery Co. ; till recently Professor 
of Mechanical Engineering in the Ohio State University ; Member 
Am. Soc. Mechanical Engineers ; Member Am. Soc. Civil 
Engineers ; Fellow of the American Association for 
the Advancement of Science ; Member Am. Soc. 
Naval Architects and Marine Engineers ; 
and Member Soc. for Promo- 
tion of Engineering 
Education. 



FIB8T EDITION. 

FIRST THOUSAND. 





JOHN WILEY & SONS. 

London : CHAPMAN & HALL, Limited. 

1896. 



T3" \-|S" 



COPYRIGHT, 
BY 
S. W. ROBINSON, 



A 



ROBERT DRUMMOND, ELECTROTYPKR AND PRINTER, NEW YORK. 



Qlo-2-1%3 



PREFACE. 



This work aims to treat the whole subject of Mechanism in 
such systematic and comprehensive way that by its aid any machine, 
however elaborate, maybe analyzed into its elementary combina- 
tions, and the character of their motions determined. 

In the classification, the System of Prof. Robert Willis has been 
followed in the main, as serving best the present purpose; and 
largely his names and terms as well. 

The work contains the substance of lectures given in my classes 
during the past twenty-seven years, with such additions and 
amplifications as might come by reason of a somewhat extended 
study of the subject not only, but as brought out in connection 
with an aptitude for the thinking out of inventions involving more 
or less novel and varied forms and combinations. 

Besides Willis, some of the authors whose works have been 
referred to with profit may be named: Prof. C. W. MacCord; 
Geo. B. Grant; J. W. M. Rankine; F. Reuleaux; J. B. Belanger; 
Ch. Laboulaye. 

Some of the topics to which special attention is invited, as em- 
bracing either entirely new solutions of important questions, or 
previously unpublished discussions and extension of inquiry con- 
cerning them, are: 

Log-spiral Multilobes as derived from one spiral; also 

Proportional Sectors. 
Easements to Angular Pitch Lines. 
Transformed Wheels. 
General Solution of Non-circular Wheels, External and 

Internal, for the case of Given Laws of Motion. 
Similar and other Wheels from Auxiliary Sectors, Plane and 
Bevel. 



IV PKEFACE. 

Special Bevel Non-circular Wheels laid out on the Normal 

Sphere. 
General Solution for Bevel Non-circular Wheels for Stated 

Laws of Motion. 
General Solution for Skew-bevel Non-circular Wheels for 

Stated Laws of Motion. 
Practical Rolling of Pitch Lines and Engagement of Tee Li-. 

in one Pair of Non-circular Wheels. 
Intermittent and Alternate Motions for Moderate or for 

High Speeds; Circular and Non-circular. 
Internal or Annular Non-circular Wheels. 
Teeth for Skew-bevel Non-circular Wheels. 
" Blocking," and Steepest Gear Teeth. 
Interference of Involute Teeth of Annular Wheels. 
Epicycloidal Engine and Accessories for Machine-made Teeth. 
Full Discussion of Olivier Spiraloids; Interference, etc. 
Cam Construction by Co-ordinates. 
Form of Roller for Cams. 

Solution for Cams with " Flat Foot " Follower. 
Cam of Constant Breadth and Given Law of Motion. 
Easements for Cams. 

Solutions for Varied Velocity-ratio in Belt Gearing. 
Non-circular Pulleys for Continuous Motion by Law. 
Solution for Cone Pulleys. 
Rolling Curve Equivalent for Link-work in General ; Plane 

and Bevel. 
Gabs and Pins for Link-work in General ; Plane and Bevel. 
Velocity-ratio in Bevel and Skew-bevel Link-work. 
A General Crank Coupling connecting Shafts in Various 

Planes and Angles. 
Practical Forms of Parts for Bevel and Skew-bevel Link- 
work. 
Varied Step Ratchet Movement. 
Face Ratchets and Clicks. 
The aim of the work is not so much to present a history of 
Mechanism, as to treat upon the principles which underlie the 
various modes of modification of motion as due to the form and 
connection of parts, and thus to enable the inventor and designer 
of machines to at once solve any problem of motion of an elemen- 
tary combination in a particular case. 



PKEFACE. V 

The treatment has been mainly by graphics instead of by analysis, 
for three reasons: 1st, because the draftsman's outfit is usually at 
hand when mechanism problems arise; 2d, because these problems 
usually do not require the precision of analysis, the graphic method 
serving for problematic work as well as for delineation; 3d, because 
analysis, though possible in a few of the simpler problems, becomes 
difficult and often impossible with very slight variations of condi- 
tions, while by the graphic method all cases, whether of simple or 
complex statement, are solved with nearly equal facility. Hence 
analysis has been employed here only to establish principles the 
application of which subsequently might be made by the graphic 
method. 

S. W. Robinson. 
Columbus, O., Sept. 26, 1896. 



CONTENTS. 



INTRODUCTION. 

Object of Principles of Mechanism , Page 

How Studied. Machine Defined. Frame ; and Trains of Mecha- 
nism. Machine Parts Classified. Elementary Combination of 
Mechanism. Motions, how Studied, 1. Synoptical Table of Ele- 
mentary Combinations of Mechanism, 2. Velocity ; Constant and 
Varied ; Angular. Velocity-ratio. Period. Cycle, 3. Revolution. 
Distinction into Driver and Follower. Directional Relation, 4. 



PART I. 
TRANSMISSION OF MOTION BY ROLLING CONTACT. 

CHAPTER I. 

Rolling Contact in General , Page 5 

Line of Contact. Axes. Plane of Axes. Point of Contact. 
Line of Centers. True Rolling Contact. Velocity-ratio in Circular 
Rolling Contact. No Slip of Surfaces in Contact, 5. Velocity- 
ratio of Non circular Arcs, 6. Variable Velocity-ratio, 7. 

Circular Wheels Page 7 

Axes Parallel. Friction Wheels. Contact between the Axes, '7. 
Contact Outside the Axes. Axes Meeting, 8. Parallelogram for 
Locating Line of Contact. Rolling Cones. Angle between Axes, 
Particular Case, 9. Axes Crossing without Meeting. Character of 
Contact between the Surfaces. Form of the Rolling Surfaces. 
Location of Line of Contact in Plan, 10. Location of Line of Con- 
- tact in Elevation. Hyperboloidal Form of Surfaces. Longitudinal 
Slip of Surfaces. Error in Early History of Case, 11. Graphic 
Construction of Skew Bevels. Proof of Construction, 12. Not 
Practical for Friction Wheels. Longitudinal Slip Determined, 13. 

Circular Intermittent Motions ... Page 13 

For Axes Parallel. Locking of Driven Wheel when Idle. 
Avoidance of Friction Clamps. Positive Locks Classified, 13. Work^ 

vii 



Till CONTENTS. 

ing Drawings of Various Circular Intermittent Motions, and Illus- 
trations of Actual Constructions of Same, 14-18. 

CHAPTER II. 

Special Non-circular Wheels , Page 19 

Complete or Incomplete. Theory of Pitch Lines. Axes Fixed. 
Three Cases : Axes Parallel, Axes Meeting, and Axes Crossing 
witbout Meeting. Notation. True Rolling Contact. Point of 
Contact on Line of Centers, 19. Radii in Pairs ; their Sum a Con- 
stant. Mutually Rolled Arcs equal Each Other. Elementary 
Sectors. Five Special Cases. Equal Log-spirals. Will Roll Mutu- 
ally and Correctly, 20. To Construct the Log-spiral Graphically. 
Geometric Series of Radii. Spiral Passing Two Given Points, 21. 
Length of Log-spiral Arc. Application of Log-spiral Wheels. Sec- 
toral Wheels. Log-spiral Levers, 22. Weighing Scales. Wire 
Cutter. Wipers. Complete Wheels, 23. Symmetrical and Uusym- 
metrical Log-spiral Wheels : By Aid of Two Spirals ; and by Angles 
and Radii, 24. Multilobed Wheels : by Reduction of Angles : by 
Assumed Angles and Limiting Radii, 25. Interchangeable Multi- 
lobed Log-spiral Wheels : as derived from One Spiral with Lobes 
Symmetrical , as derived from Two Spirals with Lobes Unsym met- 
rical, 26. Interchangeable Log-spiral Lobed Wheels. One, and 
Three Lobed Wheels, as determined from Proportional Sectors, 
21, 28. Case of Infinite Number of Lobes, 29. Easement Curves. 
Equal and Similar Ellipses. Proof, 29. Range of Velocity-ratio, 

30. Example from Centennial of 1876 of Elliptic Gear Wheels, 

31. Interchangeable Multilobed Elliptic Wheels, Holditch, 32. 
Working Drawings and Photo-process Copy. Wheel with Infinite 
Number of Lobes, 33. Equal and Similar Parabolas as Rolling 
Wheels. Equal Hyperbolas, 35, 36. Transformed Wheels. Three 
Rules, 37. Changes Characteristic of Application of Each Rule, 
38. Interchangeable Multilobes by Transformation, 39. Trans- 
formed Elliptic Wheels, 40. Transformed Parabolic Wheels. 
Wheels of Combined Transformed Sectors, 41, 42. 

CHAPTER III. 

Non-circular Wheels in General Page 44 

Three Cases : One Wheel Given to Find its Mate ; Laws of Motion 
Given to Find the Wheels ; Similar and Other Wheels from 
Auxiliary Sectors. Complete or Incomplete. Given One Wheel, 
to Find its Mate, 44. Repeated Trials. Graphic Rules for Equiva- 
lence of Lines and Arcs, 45. To Draw a Tangent to any Curve, 46. 
Application of Above Rules to Case of One Wheel Given to Find 
its Mate, 47. Examples of Above. Laws of Motion Given to Find 
the Wheels, 48, 49. Proof of Above, 50, 51. Photo-process Copies 
from Examples of Above, 52, 53. Solutions of Some Practical 



CONTENTS. IX 

Problems, in Laws of Motion Given to Find the Wheels. The 
Shaping-machine Problem, 54, 55. A Bobbin-winding Problem ; 
Solution for Conical Bobbin, 56, 57. Motion of Driver a Variable. 
Case of Multilobed Wheels. Similar and Other Wheels from 
Auxiliary Sectors, 58. Similar and Equal Multilobed Wheels. 
Dissimilar Multilobed Wheels, 59. Multilobes of Unequal Number 
of Lobes, 60, 61. 

CHAPTER IV. 

Special Bevel Non-circular Wheels Page 62 

Normal Spheres. Spherical Equiangular Spiral. One-lobed 
Spherical Equiangular Wheels, 62. Drawing on Spherical Blank 
for Pattern, 63. Two-lobed and Multilobed Bevel Non-circular 
Wheels. Interchangeable Bevel Non-circular Wheels, 64. Ellip- 
tic Bevel Wheels, 65. Multilobed Elliptic Bevel Wheels. Parabolic 
and Hyperbolic Bevel Wheels. Transformed Bevel Wheels, 66. 
Similar Bevel Multilobes. Dissimilar Interchangeable Bevel Multi- 
lobes. Partially Interchangeable Bevel Multilobes. Bevel Non-cir- 
cular Wheels of Combined Sectors, 67. Interchangeable Multilobes 
with Un symmetrical Lobes. Velocity ratio in Bevel Non-circular 
Wheels, 68. 

CHAPTER V. 

Bevel Non-circular Wheels in General Page 69 

Laws of Motion. Plane Wheels Turned into Bevels, 69. Diagram 
of Plane Wheels Possessing Correct Laws of Motion. Auxiliary 
Diagram giving Spherical Radii from Plane- wheel Radii, 70. 
Spherical Blank for Gear Pattern. One or Several Lobed, 71. Proof 
of Above, Example, 72. Laying Out Non-circular Wheels Direct on 
the Normal Sphere. Case of One Wheel Given to Find its Mate. 
Also, Laws of Motion Given to Find the Wheels. Similar Wheels. 
Lobed Wheels, etc... 73-4. 

CHAPTER VI. 

Skew-bevel Non-circular Wheels Page 75 

Illustrative Hyperboloids. Pitch Surfaces of Form of Non- 
circular Hyperboloids, 75. First Find the Plane Non-circular Wheels 
of Correct Law, which Turn into Skew Bevels, 76. Spherical Blank 
to be Worked into a Gear Pattern. Mode of Procedure, 77-79. 
Proof of Above, 80. 

CHAPTER VII. 

JNON-CIRCULAR WHEELS FOR INTERMITTENT MOTIONS Page 81 

Classification. Cases of Axes Parallel, Meeting, or Crossing 
without Meeting. Easement Spurs and Locking Arcs of Greater 
Radius than Rolling Arcs. Easement, Locking Arcs, and Teeth in 



CONTENTS. 

One Plane, with Radius of Locking Arcs Less than that of the Rolling- 
Arc, 81. Laying Out of the Pitch Curves. Photo-process Copies of 
Actual Bevel Wheels. Alternate Motion Gearing, 82. Limited Alter- 
nate Motions. With Velocity-ratio Constant, or Varying, 83. Un- 
limited Alternate Motions. Mangle Rack, 84. Mangle Wheel, 85. 
Axes Meeting, 86. 



PART II. 

TBANSMISSION OF MOTION BY SLIDING CONTACT. 

CHAPTER VIII. 

Sliding Contact in General. Page 8? 

Velocity-ratio in Sliding Contact, 87. Proof of Velocity-ratio, 88. 
Sliding and Rolling Curves with Common Law of Motion. One 
Sliding Curve Given, Find a Mate, With Law of Motion that of a 
Given Pair of Rolling Wheels, 89. Teeth of Gear Wheels, 91. 
Tracing of Sliding Curves. Odontoids, Centroids. Describing 
Templet, 92. Names, Terms, and Rules for Gear Teeth, 93. Pro- 
portions for Gear Teeth. Circumferential and Diametral Pitch, 94. 

CHAPTER IX. 

Tooth Curves for Non-circular Gearing. General Case . . . Page 95 

Axes Parallel. Templets, 95. Use of Templets in Describing 

Tooth Curves, 96. Requirements of Tooth Curves, 97. The Tooth 

Profile. Position of Tracing Point on Describing Curve, 98. The 

Trachoidal Tooth Curve, 99. Form and Size of Describing Curve, 

100. Individually Constructed Teeth. Non-circular Involute Gears, 

101. Conjugate Teeth, 102. Limited Inclination of Tooth Curves, 
103. Hooking Teeth, 104. Nearest Approach of Tooth Curve 
Toward Radius, 105. Practical Limit of Eccentricity of Pitch 
Line, 106. Eccentricity as Affected by the Tooth Profile, and the 
Addendum, 107. "Blocking" of the Gears, and as Affected by 
Length of Tooth, 108. Substitution of Pitch Line Rolling for 
Teeth iu Extreme Eccentricity. Link Substitute for Teeth, 109. 
Examples and Photo-process Copies of Actual Gears, 110, 111. 
Internal Non-circular Gears, 111. 

CHAPTER X. 

Teeth op Bevel and Skew-bevel Non-circular Wheels. . . . Page 112; 
General Case of Axes Intersecting. Describing Curve a Cone, 112. 
Tracing of Tooth Curve Direct on Sphere. Tredgold's Approximate 
Method. Involute Teeth,- 113. General Case of Axes Crossing 
without Meeting. Teeth Twisted. Correct Exact Construction not 
Possible 174, 



CONTENTS. XI 



CHAPTER XI. 

Non-circular Intermittent Motions Page 116 

Teeth Spurs and Segments. Teeth as in Non-circular Wheels. 
Engaging and Disengaging Spurs. Backlash, 116. Spurs of 
Rolling, or of Sliding Curve Form. As Adapted for High Speed, or 
Slow, 118. Rolling Spurs often Impracticable, 119. Rolling Spurs 
require Excessive Arc of Motion. Bevel and Skew-bevel Wheel 
Spurs, 120. Solid Engaging and Disengaging Segments. Forms 
such as not to "Block," 121. Undercut so as to Relieve the Shock, 
122. Counting Wheels. Alternate Motions. Limited in Move- 
ment, 123. Skips and Derangements. Spurs Employed, 124. 
Solid Segments for Reversing Motion. Best Form of Wheel for 
High Speed, 125. Axes Meeting. Normal Sphere, 126. Use of 
Motion Templets. Unlimited Alternate Motions. Mangle Wheel 
and Rack. Non-circular Form. Velocity-ratio, 127. 

CHAPTER XII. 

Teeth op Circular Gearing Page 128 

Axes Parallel. Epicycloidal, Involute, and Conjugate^ Gearing. 
Epicycloidal Gearing, 128. Some Peculiar Properties of Epicycloids 
and Hypocycloids. White's Parallel Motion, 129. Seven Styles of 
Teeth, viz.: Flanks Radial, Concave, Convex; Interchangeable Sets; 
Pin Gearing ; Rack and Pinion ; Annular Wheels. Flanks Radial. 
The Face and Flank Generated by Describing Circles, 130. Flanks 
Concave. Size of Describing Circle. Flanks Convex, 132. Inter- 
changeable Sets. " Change Wheels." Describing Circle, 133. Pin 
Gearing, 134, 135. Inside Pin Gearing. Pinion of Two Teeth, 136. 
Rack and Pinion. Wheels of Least Crowdiug, 137, 138. Rack and 
Pinion. Flanks Radial. Flauks Concave, 139. Annular Wheels. 
Peculiar Case of Interference and an Extra Contact. Wheels in 
Sets, 140, 141. Pin Annular Gearing. Involute Gearing. Curves 
Described by Log-spirals Rolled on Pitch Line, 142. Practical Mode 
of Drawing the Teeth, 143. Rack and Pinion. Annular Wheel and 
Pinion. Interference of Annular Wheel Teeth, 144. Wheel of 
Several Styles of Teeth. Conjugate Gearing. Flanks Parallel 
Straight Lines, 145. Flanks Convergent Straight Lines. Flanks 
Circle Arcs, 146. 

CHAPTER XIII. 

Practical Considerations , Page 14& 

Addenda and Clearance. Rule for Circumferential Pitch. Rule 
for Diametral Pitch, 148. To Strengthen the Teeth. Possible 
Clearing Curve. Path of Contact : Determined, Assumed, Limited, 
149, 150. Acting Part of Plank. Line of Action. Obliquity of 
Line of Action, 151. The " Blocking" Tendency. Unsymmetrical 



Xll CONTENTS. 

Teetb. Practical Construction of Tooth Curves, 152. Tooth Tem- 
plet, 153. Radius Rod for Tooth Templet. To Draw Involute 
Teeth, 154. Approximate Teeth in Practice, 155. The Templet 
Odontograph. As a Ready-made Tooth Templet. Radius Rod, 156, 
157. The Willis Odontography gives Center for Circle Arc Tooth 
Curves, 158-160. The Three Point, or Grant's Odontograph, 160, 
161. Co-ordinating the Tooth Profile, 162. Advantages of Each of 
Above Instruments, 163. 

CHAPTER XIV. 

Machine-made Teeth Page 164 

Teeth with no "Hand and Eye" Process. Epicycloidal Engine 
to Form Epicycloidal Curve, 164. Grinding the Tool to the Curve, 
165. Forming the Gear Cutting Tool, 166. The Brown & Sharpe 
Involute Cutters ; also their Epicycloidal Cutters, by Machinery of 
Pratt & Whitney Co., 167. Sang's Theory of Conjugating the 
Teeth. The Swasey Engine with Split Cutter, 168. The Grant's 
Engine with Solid Worm Cutter, 170 Form of Cutter Teeth, 171. 
The Gleason's, Corliss', and Bilgram's Gear Planing Machine. 
Stepped and Spiral Spur Gearing, 172, 173. 

CHAPTER XV. 

Circular Bevel Gearing Page 174 

Correct and Approximate Solutions. The First by Use of Cones 
with Bases formed to the Normal Spheres ; and the Second by Use 
of Normal Cones of Tredgold, 174. Complete Drawing by Tred- 
gold's Method, 175. Spiral Bevel- wheel Teeth, 176. 

CHAPTER XVI. 

Teeth for Circular Skew-bevel Gear Wheels Page 177 

Approximate Construction. A Practical, Theoretically Correct 
Solution not Known. Approximate Epicycloidal Form. Same 
Arbitrarily Dressed or "Doctored," 177. Tredgold's- Method, 
Applied on Normal Cones, 178. Amount of Error Illustrated, 180. 
Exact Solution in Olivier Spiraloids, 181. Olivier Spiraloid Explained. 
Interchaugeability of Spiraloids, 182. Interference of Spiraloid 
Teeth. Nature of Contact of Spiraloid Teeth, 183. Results of an 
Example. Principle Demonstrated. Flat Faces of Teeth when 
Remote from Gorge Circles, 184. Certain Forms Approximating 
the Olivier Gears. Olivier Worm and Gear, 187. Hiudley Gears. 
Skew-pin Gearing. Intermittent Motions, 188, 189. 

CHAPTER XVII. 

Circular Alternate Motions Page 190 

Limited Alternate Motions. Solid Engaging and Disengaging 
Parts, 190. With Attached Engaging and Disengaging Parts. The 
Mangle Wheel. Unlimited Alternate Motions, 191. 



CONTENTS. Xlll 



CHAPTER XVIII. 

Cam Movements Page 192 

Cains in General. Friction. Backlash. Solution by Co-ordinates, 
192. By Intersections. Use of Templets. Velocity-ratio, 193, 194. 
Continuously Revolving Cam, 195. Cylindric Cam, with Straight 
Path for Follower, 196. Same with Curved Path of Follower. 
Conical Cam, 197. Spherical Cam, 198. Cam Plate. Flat-footed 
Follower, with Specific Law of Motion of Follower. Uniform 
Motion, 199. Law of Motion that of Crank and Pitman. Law that 
of a Falling Body, 200, 201. Tarrying Points. Uniform Recipro- 
cating Motion. The Heddle Cam, 202. Easements on Cams. Ar- 
bitrary Easements, and those Confined to Assigned Sectors, 203. 
Case of Flat-footed Follower, 205. Cams of Constant Diameter, 206. 
Cams of Constant Breadth, 207. Cams with Several Followers. The 
Effect of Two Followers, 208. Four-motion Cam. Illustration from 
Example, 210. Duangle Cam. Return of Follower by Gravity, 211. 
Positive Return : Three Ways, 212. To Relieve Friction, 213. 
Modification of Cam to Suit Form of Follower, or Roller, 214, 215. 
Best Form of Roller. Action of Roller upon its Pin or Shoulder. 
Roller for Conic Cam. For Spherical Cam, 216, 218. 

CHAPTER XIX. 

Inverse Cams and Couplings Page 219 1 

The Pin and Slot. Form of Slot, Velocity-ratio, 219. Oldham's 
Coupling, with Shafts in Parallel. 220. Shafts not Parallel, 221. 
Peculiar Coupling. Oldham's Three-disk Coupling, 223, 224. 

CHAPTER XX. 

Escapements Page 225 

Power Escapements, 225. Anchor Escapement, 226. Pin-wheel 
Escapement, 228. Gravity Escapement, 229. Cylinder Escapements, 
231. Lever Escapement, 232. Duplex Escapement, 233. Chronom- 
eter Escapement, 234. 

PART III. 
BELT GEARING. 

CHAPTER XXI. 

Transmission op Motion by Belts and Pulleys. Rope, Strap, or 
Chain over Sectoral and Complete Pulleys. Motion 

Limited or Continuous. Velocity-ratio Varying Page 23S 

Velocity-ratio, 236. Line of Action. Non-circular and Circular 
Pulleys. Law of Perpendiculars given to Find the Wheels, 237. 



XIV CONTENTS. 

Equalizer of Gas-meter Prover. A Draw-bridge Equalizer, 239. 
Barrel and Fusee, 240. Non-circular Pulley for Rifling Machine, 
241. Example of Treadle Movement, 243. 

CHAPTER XXII. 

Circular Pulleys Page 244 

Continuous Belt. Velocity-ratio. "Slip," 244. Retaining Belt 
on Pulley. Pulley with High Center, 245. Crossed Belt. Quarter- 
twist Belt. Guide Pulley, 246. Any Position of Pulleys. Cone 
Pulleys. Problem of Cone and Counter Cone. Geometric Series of 
Speeds, 247, 251. Rope Transmission : Two Systems. Rope Belting, 
Short and Long Stretch, 251-53. For Haulage Lines. Compensating 
System. Chain and Sprocket Wheel. Teeth of Sprocket Wheels. 
Practical Application of Chains, 254-57. 

PART IV. 
LINK-WORK. 

CHAPTER XXIII. 

Rods, Levers, Bars, etc. The General Case Page 258 

Velocity-ratio, 258. Peculiar Features of Link-work Mechanism. 
Lightest-running Mechanism. Inflexible Law of Motion. Axes 
Parallel, 259. Examples : 1. Needle-bar Motion ; 2. Corliss' Valve 
Gear ; 3. Driven Piece Half the Time nearly Quiet ; 4. Small Move- 
ment in a Given Time, 260, 261. Path and Velocity of Points. 
Sliding Blocks aud Links, 261-63. 

CHAPTER XXIV. 

A Rolling Non-circular Wheel Equivalent for Every Piece of 

Link-work. Gabs and Pins Page 264 

Examples showing Several Rolling Curves, 264. Curves and 
their Linkages shown Separately, 265, 266. Rolling Curve Equiva- 
lent of Crank and Pitman. Two Cranks and Drag Link. The 
Sylvester Kite, 267, 268. Equal Cranks in Opposite Motion. Hyper- 
bolas. Elliptic Wheels, 269-71. Parabolic Wheels. Unsymmetri- 
cal Link-work and Rolling Wheels, 272, 273. Dead Points in Link- 
work. "Gab and Pin." Path of Gab and Pin. Gab and Pin in 
Practice, 274-77. Multiplying Motion by Link-work, 278. 

CHAPTER XXV. 

Conic Link-work Page 279 

Axes All Meet in a Point. Cranks Equal or not. Velocity-ratio, 
279. Rolling-wheel Equivalent of Conic Link-work. Dead Center, 
and Gab and Pin in Conic Link-work. Examples of Peculiar Move- 



CONTENTS. XV 

merits, 281. Possible Valve Motion. Hooke's Joint or Coupling, 
282-85. Almond's Coupling, 286. Crank Coupling, 287. Reuleaux 
Coupling, 288. 

CHAPTER XXVI. 

Link- work with Axes Crossing without Meeting Page 289 

Ball and Socket or Sliding Joints. Velocity-ratio, 289. The 
Willis Joint System, 290. Shafts Connected by Cranks and Angle 
Blocks, Three Examples, 291, 292. Ratchet and Click Movements. 
Running Ratchet. Varied Step Movement, 292, 293. Reversible 
Running Ratchet. Reversible Varied Rate Running Ratchet. Con- 
tinuous Running Ratchet. Forms of Teeth and Click, 294, 295. 
Friction Ratchets. Fluted Bearing. Continuous Friction Ratchet. 
Wire Feed Ratchet. Running-face Ratchet and Click, 296, 297. 



PART V. 
REDUPLICATION. 

CHAPTER XXVII. 

Pulleys or Sheaves and Ropes Page 298 

Velocity-ratio. Parallel Ropes, 298. Ropes not Parallel. Veloc- 
ity-ratio. Hay Unloader, 299. Hydraulic Elevators. Weston 
Differential Block. Velocity- ratio, 300. 



PRINCIPLES OF MECHANISM. 



INTRODUCTION. 

Ik working out the design, drawings, and specifications for a 
machine, the form, strength, and motion of the various parts 
must be determined, the last being the object of the Principles 
of Mechanism, 

In Principles of Mechanism we find the application to machines, 
of the principles of Kinematics, or Cinematics, the elementary 
combinations of mechanism of which machines, being studied 
separately. 

A Machine may be defined as a combination of fixed and mov- 
ing parts or devices, so disposed and connected as to transmit or 
modify force and motion for securing some useful result. 

The fixed parts constitute the frame or supports for the moving 
parts. 

The moving parts constitute a train or trains of mechanism. 

A train of mechanism may be primary or secondary; the former 
being supported directly by the frame, and the latter by other 
moving parts. 

All the moving parts of machines may be regarded as mechani- 
cal devices and classified as follows: 

1st. Revolving shafts. 

2d. Revolving wheels or cams, with or without teeth. 

3d. Rods or bars with reciprocating or vibratory motion, or 
both. 

4th. Flexible connectors depending on friction. 

5th. Flexible connectors independent of friction. 

6th. A column of fluid in a pipe. 

Trains of mechanism consist of combinations of the above* 
devices, the least number securing a modification of force or motion 
in a given case, being an elementary combination. 

The study of the motions of a machine is usually pursued by 



6 PRINCIPLES OF MECHANISM. 

taking up separately the elementary combinations of mechanism 
composing the machine. 

Professor Robert Willis of Cambridge, England, was the first to 
present a thorough, systematic, and comprehensive table of the 
elementary combinations of all mechanism. In our study of 
mechanism to include all kinds and varieties without omissions, 
Ave can do no better than to follow this table, as below : 



SYNOPTICAL TABLE OF THE ELEMENTARY COMBINATIONS 

OF MECHANISM. (Willis.) , 



Mode of 


Directional Relation Constant. 


Directional Relation 
Changing Period icklly. 


Motion. 


Velocity-ratio 
Constant. 


Velocity-ratio 
Varying. 


Veteefty-ratio 
Constajft or Varying. 


By rolling con- 
tact. 


Rolling cylin- 
ders, cones, and 
hyperboloids; 
pitch-circles of 
circular gear - 
wheels and sec- 
tors. 


Pitch-lines of non- 
circular gear- 
ing, complete or 
sectoral. 


Pitch -lines for 
mangle - wheels 
and mangle- 
racks; limited al- 
ternate motions. 


By sliding con- 
tact. 


Tooth - curves; 
segmental cams; 
screws; worm 
gearing. 


Tooth-curves for 
noncirculargear- 
ing ; cams ; pin 
and slotted le- 
ver ; irregular 
worm gearing ; 
stop motions. 


Cams in general; 
pin and slotted 
lever ; double 
screw ; swash- 
plate ; escape- 
ments. 


By wrapping con- 
nectors, or belt 
gearing. 


Band or belt and 
pulleys ; chain 
and sprocket- 
wheel. 


Cam-shaped pul- 
ley and belt; 
fusee and chain 
or chord. 


Cam - shaped pul- 
ley and lever, or 
tightener ; trea- 
dle motion. 


By link-work. 


Equal cranks with 
link; lever with 
proportional 
links;bell-crank 
and links. 


Equal cranks with 
link ; unequal 
cranks with 
link ; Hooke's 
universal joint. 


Crank or eccentric 
and pitman; 
ratchet- wheels 
and clicks; un- 
equal cranks and 
link. 


By reduplication. 


Cord and pulley; 
pulley - blocks 
with parallel 
ropes or chains. 


Pulleys, with rope 
or chain not par- 
allel. 





NAMES AND TERMS. 

In the study of mechanism certain terms are used, some of 

which are defined below : 

Velocity. — Time is required for a point to move a distance along 
a line or path. For uniform motion the distance moved over per 
unit of time is the velocity, usually considered as feet per second. 

Thus velocity is the rate of motion, or movement. 

For the case of uniform motion, any distance passed over in a 
corresponding given time, divided by that time, gives the velocity. 

When the velocity is not constant, the space passed over in a 
very short time is divided by that short time ; as, for instance 
0.3 ft. in 0.1 sec, when the velocity will be 

— = 3 ft. per sec, 

which is the velocity or rate of moving at the instant considered. 

In mathematical calculations the space and time are often 
reduced to infinitesimals for extreme exactness, but this is rarely 
necessary in the study of mechanism. 

Angular Velocity. — Velocity may here be distinguished as linear 
and angular ; the former being the rate of motion along any line, 
while the latter is the rate of motion of a point at a distance unity 
from a center of angular motion. Either may be constant or 
variable. If variable, the rate is to be expressed for an instant 
by taking the ratio of an element of space to the corresponding 
element of time. 

Velocity-ratio. — This is the ratio of two angular or two linear 
velocities; or, in some cases, of an angular and a linear velocity. 

Period. — Moving parts of machines continue in the repetition 
of certain definite complete movements. 

The time for one of these complete movements may be called a 
period, though this term is but little used in mechanism. 

Cycle. — The time for several moving pieces, considered collec- 
tively, to return to their given initial positions may be called a 

3 



4 PKINC1PLES OF MECHANISM. 

cycle, though this term is of comparatively little use in mechanical 
movements. 

Revolution. — A revolution is to be considered a complete turn., 
while the term rotation may be applied to any portion of a turn. 

Driver and Follower. — Ln any elementary combination of mech- 
anism one piece always drives the other, the one therefore heing 
called the driver, and the other the follower. 

Directional Relation. — This term has reference to the relation 
of the directions of motion of driver and follower. When one 
never reverses its direction of motion unless the other does also, the 
directional relation is constant, otherwise it is said to be variable. 



PART I. 

TRANSMISSION OF MOTION BY BOILING 
CONTACT 



CHAPTER I. 
ROLLING CONTACT IN GENERAL. 

Rolling Contact in elementary combinations of mechanism im- 
plies a driver which turns a follower by rolling against it without 
slipping, the driver and follower having axes of motion which are 
at a constant distance apart. 

The Line of Contact of thick pieces is always a straight line like 
that of the contact of a pair of parallel straight cylinders tangent 
to each other. 

Sometimes reference is had to the plane of the axes, or plane 
which contains both axes. 

The line of contact of thick pieces in true rolling contact will 
evidently remain in the plane of the axes, and in transverse sections 
the point of contact or of tangency of the rolling curves of section 
must likewise remain on the line joining the centers of motion, 
called line of centers. 

It is evident that in true rolling contact no slipping can occur, 
and that arcs which have rolled correctly in mutual contact must 
be equal to each other. 

VELOCITY-KATIO IN ROLLING CONTACT. 

First. For Circular Arcs and Cylinders. — Suppose Fig. 1 to rep- 
resent a transverse section of a pair of rolling circular cylinders 
with axes at A and B. The line of centers is AB, and C on that 
line is the point of contact. Take Cb = Ca, and draw the radii 
R and r. Now if Ca rolls without slipping on Cb, a and b 

5 



PKItfCIPLES OF MECHANISM. 



will eventually come into contact at <7on the line of centers, when 

R will have the position AC, and 
r the position BC; while the 
axis A will have turned through 
the angle CAa, and the axis B 
through the angle CBb. The 
point e will describe the arc eel, 
and the point g the arc gf. If 
Fig. 1. these movements occur in one 

second of time with uniform motion, and if e and /are taken at a 

units distance from A and B respectively, then 




and 
Also 

and 

Whence 

since 



V. 



eel = angular velocity of A 
gf = angular velocity of B 
Ae: Mr.de: Car. V: Ca, 
Bg : r r.fg : Cb r. v : Cb or Ca; 
Vx R = Ca x Ae = v Xr = Cb X Bg. 
V_ r _BC 
v ~ R~AC 
Ca = Cb, and Ae — 1 = Bg, 

which is the velocity ratio of the axes A and B for motion trans- 
mitted from A to B by the arc Ca rolling on the arc Cb, and equals 
the inverse ratio of the radii R and r. 

As this is true of any transverse section of the cylinders, it is 
true of the entire cylinders. Hence the important principle, viz.: 
the velocity-ratio of truly rolling cylinders is equal to the inverse 
ratio of the radii of those cylinders. 

Second. For Non-circular Arcs and Cylinders. — Let Fig. 2 
represent a transverse section of a pair of correctly rolling non-cir- 
cular cylinders, or portions of 
them, with A and B as axes. 
Then AB is the line of centers, A 
C the point of contact, aAc 
and bBd the non-circular sec- 
tors. Draw circular arcs eCg 
and fCli, and the velocities V 
and v as shown. Then V is common to aAc and eAg. Also v is 
common to bBd and fBh, and the velocity-ratio is the same for 
the non-circnlar as for the circular sectors, the angles to the sectors 
being regarded as very small. 




Fig. 2. 



CIRCULAR ROLLING WHEELS. 



The segments of the line of centers being the same for the non~ 
circular as for the circular sectors, we have 

V BG 



Velocity-ratio 



AG y 



that is: the velocity -ratio in non-circular ivlieels at any instant' is 
equal to the inverse ratio of the segments of the line of centers. 

As these segments are continually varying in non-circular 
wheels, it follows that the velocity-ratio for such wheels is also 
variable. 

CIRCULAR ROLLING WHEELS. 
DIRECTIONAL RELATION CONSTANT. VELOCITY -RATIO CONSTANT. 

Friction Wheels and Pitch Lines for Circular Gearing. 

For this subject the wheels are circular, with velocity-ratio con- 
stant; and we have three general cases. 

I. Axes Parallel. For this case there are two subdivisions. 
First : Contact between the axes as in Fig. 3. Second : Contact 
outside the axes as in Fig. 4. 




Fig. 4. 



These friction wheels, or pitch lines, as the case may be, are so- 
simple in the theory of mechanism that no farther treatment seems 
necessary, unless the following graphic j)roblems are considered. 

First. Contact Between the Axes. — The axes being at a fixed 
distance apart and parallel, let them be represented in Fig. 5 bv the 
lines at A and B. Take the velocity-ratio as being given with the 
value 2/5. Then 

V_2 _ r 
v ~~ 5 "~~ R' 



3 



PRINCIPLES OF MECHANISM. 




Fig. 5. 



As the sum of the radii equals the distance between the axes, 
then if we take r = 2 inches and R — 5 
inches, and add, we have 7 inches. If 
this be less than the distance between 
the axes, double each and add, or triple 
each and add, until the sum is greater 
than the distance between the axes, as 
AB, Fig. 5. Then with AB as a radius in the dividers place one 
point at A and strike the arc intersecting at B. Draw a straight 
•line AB, and lay off on this line from A the greater part of the 
rsum AC, and from B the lesser part, giving the point C. Through 
■C draw the line of contact. Then the perpendicular distances 
from the axes A and B upon the line of contact will be the radii of 
the wheels, which will be in simple proportion with the distances 
BC and AC. 

Second. Contact Outside the Axes. — Here also we take A 
and B for the axes, and C the line of contact, the three lines being 




Fig. 6. 



With the same example of velocity-ratio 
as before, take the difference of the values 
b and 2, or 3 ; or twice or more times these 
values if 3 is too small. Then with the 
dividers opened to the difference found, 
place one point of the dividers at A and strike an arc intersecting 

at B, and through AB draw the 
straight line A C, extended as far as 
necessary. Then with B as a center 
and the lesser of the above values lay 
off BC; and the greater should equal 
A C Now drawing the line of con- 
tact through C, parallel to A and B, 
we have the radii as perpendiculars 
from A upon C, and from B upon C. 
II. Axes Meeting. Take AO and 
BO as the axes, meeting at 0. The 
diameter of the wheel A may evident- 
ly be any line CAE, perpendicular 
to its axis AO, and the radius AC = 
R. Then the diameter of the mat- 
ing wheel must be CF, and the radius BC = r. 

To determine the relation between the radii of the wheels and 




CIRCULAR ROLLING WHEELS. 



"their angular velocities, lay off the velocity of A on the axis of A, 
and the velocity of B on the axis of B, as On and Ob respectively, 
and complete the parallelogram Oacb by drawing ac and be. Also 
draw cd and ce perpendicular to the axes, and we have the triangle 
acd similar to bee, the angle cad being equal to cbe. Then 



V:v :: Oa : Ob 
Whence the velocity-ratio 



:bc 


ac : 


: ce 


: dc 


: r 


V _ 

V 


r 









B. 



proving the correctness of the parallelogram for conveniently locat- 
ing the line of contact when the velocity-ratio and angle between 
the axes are given. 

If another pair of circles tangent to each other be drawn on the 
axes A and B, as, for instance, those tangent at the point C on 
the line OC, the ratio of the radii of this pair will be the same as 
that of the circles tangent at C, as is plain from the geometry of 
the figure. 

Hence correctly rolling surfaces for this case must be cones 
with their vertices in common as at 0. 

In the above the contact is between the axes, which are situated 
to intersect at an angle of less than 90 degrees. 

When the angle between the axes is greater than 90 degrees 
there seems to be a second case, as if the contact were outside this 
angle as shown in Fig. 8. But 
the principles are all the same, 
and the wheels are readily made 
in practice. 

The wheel A of Fig. 7 is shown 
at A' , Fig. 8, so that the cone B , 
may have a mate AG or A' C on 
the axis A. 

A peculiar result is obtained 
when the line OC is perpendicular 
to A 0, the wheel A being a plane 
circular disk instead of a cone. 
But 0, the intersection of the 
axes, is the common vertex of all 
the rolling surfaces AA'B, etc., in every instance of correct rolling 
contact. 




Pig. 8. 



10 



PRINCIPLES OF MECHANISM. 



Some old mill gearing made wholly of wood ignores the above 
conic forms of the theoretic surfaces, where one wheel has teeth 
upon its side and the other upon its edge. 

III. Axes Crossing Without Meeting. — For convenience, take the 
vertical projections of the axes and the line of contact of the roll- 
ing surfaces as here shown, parallel to the " ground line " of the figure 
This line of contact must be a straight line, and an element of each 
of the surfaces. 

If the surfaces have a possible existence for this peculiar case, 
it is plain that the form of each may be conceived to be described 

Fig. 9. 




Fig. 10. 



by imagining the line of contact to be fixed to the one axis and to 
be revolved about it, thus generating that surface, first for the one, 
and then for the other. The form of surface thus described is the 
liyperooloid of revolution with axes A A' and BB\ Fig. 9. The 
smallest diameters will be at the common perpendicular, or short- 
est distance between the axes A' 'B' ', which perpendicular is inter- 
sected at 0' by the line of contact C'O'C. 

To locate the line of contact in the horizontal projection, imag- 
ine the surfaces to be extended to infinity, where they become roll- 
ing cones. Thus we can locate the line of contact in Fig. 10'to 



CIRCULAR ROLLING WHEELS. 11 

correspond with the velocities V and v as in conic wheels, Fig. 7, 
laying off Fon the axis of A, and v on the axis of B, as shown, and 
completing the parallelogram with GC, the diagonal on the line of 
contact in Fig. 10 when sufficiently extended. In the drawing, it 
is most convenient to place the line of contact CO A r parallel to the 
ground line, as in Fig. 10. 

To locate the line of contact in Fig. 9: First, it is plain that it 
will be parallel to the axes in order to meet the case of rolling 
cones of infinite extent. Second, to find the point 0' which di- 
vides the common perpendicular A'B' into segments, or gorge-cir- 
cle radii/draw A' C and B'G' parallel to the.respective axes A and B 
of Fig. 10. Through the intersection C ', Fig. 9, and parallel to the 
axes draw the line of contact. Thus all the essential lines of Figs. 
9 and 10 become located, and the hyperboloids can be drawn in as 
shown, using the axes A and B, Fig. 10, as geometric axes, and the 
line of contact OGJSf as an asymptote for the hyperbolic curves. 
Afterwards Fig. 9 can be completed by the theory of projections. 

The correctness of this construction for the triangle A' B'C is 
shown by aid of the end view at the left of Fig. 10, obtained by 
passing a plane, FG, perpendicular to the line of contact, and re- 
volving it to 1JMN, in which some point K is where the line of 
contact pierces this plane. In this cross-section, the point K is the 
point of contact of the rolling hyperboloidal surfaces as intersected 
by the plane, in curves tangent at K, as shown. As /and <7are 
centers of rolling motion in this plane, with IK and JK as radii, 
K must be on a straight line from I to J, which locates the line of 
contact as between A r and M, or between A' and B f as well; since 
the line of contact KG, O' G' is parallel to the projection of the 
axes in Fig. 9. But the triangle A'B' G f is similar to FGO, and 
INK to JMK, so that A' O' + B'O' = NK '-=- MK, proving true 
the construction Fig. 9 for the position of C'O f . 

It may be noted that a system of hyperboloids adapted for roll- 
ing upon each other in pairs interchangeably, may be drawn by 
making K, CO the line of contact, the point of intersection of all 
the axes in plan, IJ the locus of all their intersections with the 
plane FG, all axes being parallel in their projections Fig. 9. 

A notable point in the history of this case is that the early 
writers, from analogy with the case of bevel wheels, assumed that 
the radii R' and r' were in the simple inverse ratio of the. angular 
velocities. The French author Belanger, however, gave an ex- 
tended and rigorous proof to the contrary, agreeing with the above 



12 PRINCIPLES OF MECHANISM. 

graphical construction A'B'C of Professor Rankine. To show 
this construction to be correct, suppose the wheels to make a small 
turn in rolling contact so that the line of contact, OC produced, 
makes a small displacement from OC to ab, Fig. 10. Then Oa 
may be regarded as the horizontal projection of the corresponding 
movement of the point 0' in the circle A'O'. Likewise Ob may be 
the plan of the corresponding movement of 0' in the circle B'O'. 
But a and b should be in a line parallel to CO, as required for 
no lateral slipping of the surfaces; longitudinal slipping being nec- 
essarily allowed to provide for the endwise sliding of the teeth of 
these wheels while engaged, and in action, the teeth being laid out 
to coincide, in position, with a series of lines of contact, or straight 
elements of surface properly distributed around the hyperboloids 
as explained under Teeth of Wheels. 

Thus conditioned, we see that the triangle Odb is similar to 
A'B'C 

To prove this construction correct we have from Fig. 10 

B' V = Oa, r'v = Ob, Oa cos aOd --= Od = Ob cos bOd, 

and from the parallelogram OC we have 

V sin eOC = V sin aOd = v sin fOC = v sin bOd, 

whence 

R' _ v . Oa v cos bOd _ sin aOd cos bOd _ tan aOd 
T 7 "~ V. Ob ~~ V cos aOd ~ snT&W X cos aOd ~ tan bOd' 

^relation that must exist between R',r', v,and V, as consistent with 
the allowed longitudinal sliding along ab, but not lateral slipping 
of surfaces. 

From Fig. 9, as constructed, we obtain 

O'C tan A'C'O' = R' 9 O'C tan B'C'O' = r' y 

-whence 

R' __ tan A'C'O' tan aOd 
~V ~ tan B' C r 6' ~ tan bOd' 

the same relation as obtained from Fig. 10 as necessarily con- 
structed. Hence the construction for A'O'B' as described is the 
correct one. 



CIRCULAR ROLLING WHEELS. 13 

These wheels, sometimes called rolling hyperboloids, can never 
serve as practical rolling surfaces or friction wheels, from the fact 
that the unavoidable longitudinal sliding induces lateral slip; bat 
in skew-bevel gearing they are entirely practical. To determine 
the longitudinal slip, compare ab with Oa or Ob, these being cor- 
responding values of this slip, the movement of the end of R', or 
of ?•', respectively. 

INTERMITTENT MOTIONS. 

These might have been classified with respect to the relation of 
axes, as parallel ; meeting; or not parallel and not meeting; but it is 
believed advisable to classify them with regard to their peculiarities, 
and treat them here together. 

TJiis name is given to movements in rolling contact where the 
driver A, Fig. 11, moves continuously while the follower, or driven 
piece B, has periods of rest and motion. 

The class here considered, of velocity-ratio constant, includes 
such of these movements as have circular rolling arcs, here dotted 
as pitch lines for the teeth. 

To be thoroughly practical, the driven piece should be positively 
locked when idle, and be stopped and started in gradation of 
motion to avoid shocks and breakage. 

Spring checks, friction clamps, etc., should never be relied 
upon, as a momentary change in speed may prevent the spring 
from properly catching in its notch, or the friction clamp from 
wear may get out of adjustment and fail to act. Serious breakages 
have been known to occur in machines for making wire hooks for 
hooks and eyes, where this movement has been used with a friction 
clamp instead of positive locks. 

These movements possessing positive locks are made in at least 
two ways which are thoroughly practicable. 

First, with the locking arc of greater radius than the rolling 
arc, and provided with easement spurs. 

Second, with the locking arc of smaller radius than the rolling- 
arc, and provided with easement segments. 

First. Here the locking arc JK on the wheel A, Figs. 11 and 12, 
is given a larger radius than the rolling arc DHE, in order that 
the curves, EJ and DE, of varying radius may act upon the saddle 
curve EG while the first point E or the rolling arc of B is moving 
into engagement with the first point of the rolling arc EH, to 
prevent the wheel B from moving out of place in the one direction,. 



14 



PRINCIPLES OF MECHANISM. 



the spurs L and N preventing derangement in the other direction. 
The spurs L and N are on the front side of the wheels, while the 
spurs P and R are on the opposite sides, and dotted. 




Fig. 11. 

In Fig. 11 the pin N moves toward the spur L, fastened upon 
B, striking it first as the locking arc JGK begins to fall away, the 
latter being cut away more and more toward E so that B is just 
allowed to turn with but slight backlash until the pin noting against 
the spur has thus brought the initial point of the rolling arc FI to 
contact with the initial point of the rolling arc EH, when rolling 
action begins and the spurs L and N go out of use for that revolu- 
tion. When the rolling has continued through H to D, the lock- 
ing arcs and spurs again come into action until B is locked again 
in the position shown, it having made one revolution while A 



CIRCULAR ftOLLING WHEELS. 



15 



made but part of one, B remaining idle and locked while A com- 
pletes its revolution. 

It will be seen that the pin strikes the spur quite abruptly in 
Fig. 11 unless the spur is considerably inclined, ani starts B sud- 




Fm. 12. 



denly, so that in high speed the pin or spur may be broken, while 
in Fig. 12 the two spurs, LM and NO, extending nearly to the 
axis of motion of the driver A, start B slowly at first and with 
greatly reduced shock, but as the contact between the spurs moves 
outward, the wheel B is accelerated until F meets E when rolling 
contact begins. By placing a pair of spurs on each side of the 
wheels, they are adapted for running either way; or if in one direc- 
tion only, it is important to have one pair for accelerating B at 
starting, and another for retarding it gradually while approaching 



16 



PEINCIPLES OF MECHANISM. 



the locking arc. Thus the length of the spurs may be fixed in a 
particular case by the judgment of the designer. 

The rolling-circle arcs of these wheels are what bring them 
under the present topic of rolling contact and constant velocity- 
ratio. The spurs will usually work by sliding contact, though 
they may act by rolling, but with velocity-ratio varying, and their 
forms may be studied under their proper headings. (See Fig. 141.) 

Second: with locking arc smaller in radius than the rolling, 
arc, as in Fig. 13, where B is shown as approaching the locked 




Fig. 13. 



position, and with H and I the rolling arcs. As E approaches F it 
glances and slides on F in such a way as to start B slowly, and 
with acceleration until the initial points of the rolling arcs H and 
I come to contact when rolling begins with velocity-ratio constant 



CIRCULAR ROLLING WHEELS. 17 

through the rolling circular arcs. When D approaches G, the 
reverse action occurs until B is again locked on the locking arc as 
shown to soon occur. 

In the details of these wheels, the locking arcs are cut away to 
some extent, as shown near E and D, to allow the points F and G 
to turn while passing from lock into action, and B is cut away 
somewhat in the lock between F and G. Also the circular rolling 
arcs H and / have easements at their ends, thus introducing short 
non-circular rolling arcs at E and D, to be set with gear teeth, when 
they are laid out for the rolling arcs treated as " pitch lines " for 
teeth. 

Fig. 13 has the advantage of simplicity, A and B being each in 
practice cast in one piece, while Figs. 11 and 12 require spurs, 
pins, etc., which usually must be made and put on with screws or 





Fig. 15. 
Fig. 14. 

rivets. The form Fig. 13 is used on certain binders of reaping 
machines. 

In these figures there is but one locking arc, or lock, to each 
wheel, so that A makes as many revolutions in a given time as B. 
But it is readily seen that A may have several rolling and locking 
arcs in alternation in a circumference; likewise B may have several 
rolling arcs and locks in its circumference, and the number differ- 
ing from that of A, as, for instance, A may have three and B two. 

These wheels for intermittent motion may be made with axes 
meeting, or as not parallel nor meeting, under the principles for 
those cases as already laid down except for the easement spurs, and 



18 



PRINCIPLES OF MECHANISM. 



segments, which, whether acting by rolling or sliding contact, may 
be treated by principles considered later, which see. 

In practice, the rolling arcs in the above intermittent wheels 
must be set with gear teeth as above mentioned, the proper con- 
struction of which teeth will be considered later under Teeth of 
Wheels. 

To give a clearer idea of these wheels when complete with 

teeth, locks, spurs, easements, etc., photo-process copies are here 

.-'*\ given, where Fig. 14 is to represent 

Fig. 12 above, as complete and ready 

for action. The rolling arcs of Fig. 12 

are here provided with teeth. 

In Fig. 15 is illustrated a pair of 
wheels explained in Fig. 13, where the 
circular rolling arcs are terminated 
with short portions of non-circular 
arcs or easements, the teeth being 
extended over both. The distance 
between centers is about 4-^ inches. 
In Fig. 16 we have the same as in 
FlG - 16 - Fig. 15 turned into bevel wheels with 

axes meeting at an angle of 30 degrees, though the angle may 
be 90 degrees more or less. 

CIRCULAR ALTERNATE MOTIONS. 

The few examples for this case are deferred to page 82. 




CHAPTER II. 
ROLLING CONTACT OF NON-CIRCULAR WHEELS. 

TELOCITY RATIO VARYING.— PITCH LINES OF NON-CIRCULAR 

GEARS. 

These may be complete, that is, filling the full circle of 360 de~ 
grees, so as to admit of continued rotation, one revolution after an- 
other, or they may be incomplete or sectoral. 

This subject may very properly be regarded as the general 
theory of pitch lines of gearing; and as it is a subject of much im- 
portance it will be treated at some length. 

The axes of motion are supposed to be fixed in position, and^ 
as in circular wheels, there are three cases, viz.: axes parallel; axes 
meeting; and axes not parallel and not meeting. 

Case I. Axes Parallel. 

Here, as in circular wheels, the contact point may be between 
or outside the axes. 

The following notation will be convenient for use : 
A = the driving wheel, or axis. 
B = the driven wheel, or axis. 
C = the point of contact of wheels. 
c = the distance between axes. 
R = the radius of wheel A, and = AC. 
r = the radius of wheel B, and = BG. 
V= angular velocity of wheel A. 
v = angular velocity of wheel B. 
8 = stated, or definite, portion of arc of A. 
s = stated, or definite, portion of arc of B. 
The essential conditions for true rolling contact of these wheels 
are: 

First. The axes must be fixed at a distance c apart. 
Second. The point of contact must remain on the line of 
centers. 

19 



20 PRINCIPLES OF MECHANISM. 

Third. The sum of any pair of radii must equal a constant, 
or R -\- r = c = const. 

Fourth. Arcs having rolled truly in mutual contact are equal, 

or S — s. 

For convenience in the study of these wheels, both A and B 
may be regarded as consisting of elementary or small sectors and 
be treated in mating pairs, with limiting radii in pairs, and arcs in 
pairs, etc. Thus a pair of radii will always be R + r — c, and a 
pair of arcs will be S = s, the capitals belonging to driver A, and 
the small letters to driven wheel B. 

Special Cases of Non- Circular Wheels. — There are but five 
simple and readily demonstrated cases of correct-working non- 
circular wheels, viz. : 

First. A Pair of Logarithmic Spirals of the Same Obliquity. 
(Sometimes called Equiangular Spirals.) 

Second. A Pair of Equal and Similar Ellipses. 

Third. A Pair of Equal and Similar Parabolas. 

Fourth. A Pair of Equal and Similar Hyperbolas. 

Fifth. Transformed Wlieels. 

First. Equal Logarithmic Spirals. 

The leading characteristic of the logarithmic spiral is that it 

has a constant obliquity through- 
out. 

Taking A, Fig. 17, as the ori- 
gin, or pole, of the spiral A, the 
angle A CD, between the radius 
■^ vector AC and normal CD," is 
called the obliquity of the spiral, 
its value being the same for all 
points of this spiral. If a second 
F 1 ®- !?• spiral, B, of the same obliquity be 

placed in contact with the first, as shown in Fig. 17, the angle 
BCE being equal to the angle ACD, the contact point C will be 
found to be situated on the straight line AB, joining the poles, for 
whatever point of contact between the spirals. 

Then taking the points, or poles A and B as fixed axes of 
motion, the spirals will roll upon each other without slipping, C 




ROLLING CONTACT OF NON-CIRCULAR WHEELS. 



21 



being the point of contact and AB the line of centers, and the con- 
ditions for rolling contact of non-circular wheels are satisfied. 

TO CONSTRUCT THE LOGARITHMIC SPIRAL. 

In the logarithmic spiral, for equal angles about the pole, the 
radii vectors are in geometric progression. Draw two straight lines 




Fig. 18. 

from 0, Fig. 18, and a line ah and kb. Then draw bl parallel to 

ah, Ic parallel to kb, cm parallel to lb, etc., as far as desired. Then 

it is found that the lines Oa, Ob, Oc, Od, etc., are in geometrical 

progression, or form a geometric series. The same is true of Oj, 

Oh, 01, etc. ; or of aj, bh, cl, etc. Hence any one of these systems 

of lines may be used for radii vectores of the spiral. Through 0, 

Fig. 19, draw a series of lines at 

equal intervening angles, and lay 

off anyone of the above series of 

lines as Oa, Ob, Oc, etc., and trace 

the curve abc, etc., through the 

points thus determined. This 

curve is a logarithmic spiral, the 

construction making it evident that Fig. 19. 

the obliquity is constant. 

It is difficult to construct a spiral, as above, that will pass 
through two given located points as a and c in Fig 20. For this 
case draw a semicircle to a diameter Aa -\- Ac 
as in Fig. 21, and the perpendicular Ab will be 
the radius which bisects the angle a Ac, Fig. 20, 
thus determining the point b in 
the curve abc. Then a similar 
construction of Fig. 21 for the 
A la sector aAb will give the radius 
Fig. 21. Ad : and another construction 






22 



PRINCIPLES OF MECHANISM. 




the radius ae, etc., in continual bisection of the angles until the 
desired number of points in the spiral ac 
3 is obtained. 

LENGTH OP THE LOG-SPIRAL ARC, 

Taking as the pole, OB and OE as the 
limiting radii vectores of the log-spiral sec- 
tor OBE, draw a tangent EF at E. Make 
OG = OB, and draw GF perpendicular to 
OE. Then EF= the arc DE of the spiral. 
For a small portion dEg this appears evi- 
dent, as dE and fE are equal; and the same 
at the limit of every short arc. From this it 
appears that EH= the arc of the spiral 
H from the pole 0, to E, OH being perpen- 
Fig. 22. dicular to OE. 

APPLICATIONS. 

SECTORAL LOG-SPIRAL WHEELS. 

Fig. 23 shows a mating pair of log-spiral sectoral wheels, where 
the arcs and also the differences of 
the limiting radii of the sectors are 
equal; AB being equal the sum of 
a 'pair of radii. 

The velocity-ratio for the posi- 
tion of the wheels shown in the 
V_BG_ r 
v~ AC~ R' 

E/l R and r being the pair of radii to 
the point C. 

LOG-SPIRAL LEVERS. 

Fig. 24 illustrates the application 
of log-spiral levers, where a rod D 
is thrust forward or back by mov- 
ing handle E, the levers working by 
Fig. 24. rolling at C. 

Another application is shown in Fig. 25 to a wire-cutter, 
which in practice has been found to work well. 

A possible application to weighing-scales is illustrated in 
Fig. 26. 



figure is 





ROLLING CONTACT OF isOitf-CIRCULAR WHEELS. 23 

In Fig. 27 is shown a pair of " wipers " in steam-engine valve- 





Fig. 26. 

gears of the log-spiral order. Here the center B is at an infinite 

distance away because the driven spiral moves 

in a straight line, being supported on a rod 

sliding in guide-bearings DE. The spiral B 

in this case becomes a straight line as shown. 

The velocity-ratio is that of the linear 

velocity of sliding of B, and of the angular 

velocity of A, or V; and we have linear 

velocity of B = AC. V, or the 

, ., .. linear velocity of B .„ 
velocity-ratio = ^ — = AC. Fig. 27. ' 

We note that the velocity of B is the same as that of the point O 
as revolving about A ; since linear velocity of B = AC X V. 

COMPLETE LOG-SPIRAL WHEELS. 

First. Symmetrical Unilobed Wlieels. — In this case each wheel is 
composed of two equal sectors of the spiral, each sector being one 
of 180 degrees, as shown. These admit of continuous rotation of 
A and B with a variable velocity-ratio, for which, at any instant, 

V _BC 

v~AC' 



If the velocity of A is constant, that of B will be variable. 



24 



PKINCIPLES OF MECHANISM. 



Wheels like these are termed lobed wheels, because extended out 
to one side in a lobe. The wheels in Fig. 28 are unilobed. 




Second. Unsymmetrical Unilobed Wheels. — Herb each wheel 
consists of a pair of unequal sectors as in Fig. 29. 




Fig. 29. 

1. By aid of two log-spirals. In this case construct one log- 
spiral as in Fig. 19, say right-handed, and a second one, left- 
handed, on a transparent paper. By placing one over the other, 

with poles coincident, one may be 
turned around on the other until the 
desired size is seen lying between two 
consecutive intersections, as in Fig. 
30. Thus ADE may be chosen, or 
AEF. If the first is too small and 
the second too large the transparent 
tracing may be turned to another 
position over the other, changing the 
wheels to the desired size. 

2. By assumed angles and radii. 
Here proceed as by the process of 
Fig. 30. Figs. 20 and 21, where the radius Ab 

divides the angle a Ac and is made a mean proportional between Aa 




Logarithmic spiral wheels. 25 

and Ac, as in Fig. 21. Again, find a mean proportional between 
Aa and Ab for a radius equally dividing the angle aAb, etc., etc., 
ior as many points in ab as desired. Likewise for the arc be. 

Third. Multilobed Wheels. — 1. By reduction of angles. Sup- 
pose a series of equidistant radii be drawn in Fig. 28 or 29, and that 
the angles thus found be reduced by one half. Fig. 29 would then 
be changed to Fig. 31, giving the half of a 2-lobed wheel ; the 




other half being like it. This wheel would work correctly with 
another like it, making a pair of 2-lobed wheels. 

Again, by reducing the angles of Fig. 29 to a third their value, 
the third of a 3-lobed wheel would be obtained, from which the full 
wheel could be obtained by uniting three copied sectors. Likewise 
a 4-lobed, 5-lobed, etc., wheel could be produced. 

These wheels would work together in pairs of equal lobes, but 
not interchangeably. 

2. By assumed angles and limiting radii. In -Fig. 31 lay off 
AE — AF as maximum radii, and AD as a minimum radius. 
Bisect the angle DAFhj AG and find the length of AG as in Fig. 
21, thus obtaining a point G in the arc DGF. Similarly bisect 
GAF by AH, find AH as by Fig. 21, giving H a point in the curve. 
The point I is found in the same way, and other points between, to 
the extent desired, when the curve FHD can be drawn in. 

Likewise for sector DAE, giving the half of a 2-lobed wheel 
from which the whole wheel is readily formed either by copying 
the second half from Fig. 31 as rights and lefts with this half, 
making the wheel symmetrical with respect to EF as an axis of 
symmetry, or by swinging the copied half around on the paper 
180° and joining it to this half EDF, giving a non-symmetrical 
wheel. 

Pairs of 3-lobed, 4-lobed, etc., wheels may be thus produced, but 
they would not work interchangeably. 



26 



PRINCIPLES OF MECHANISM. 



INTERCHANGEABLE MULTILOBED LOG-SPIRAL WHEELS. 

1. As derived from one spiral. 
A 1-lobed wheel as in Fig. 28 will 
consist of two equal sectors of 180° each, 
as DAGH, Fig 32, or I AGE, etc., ac- 
cording to size. 

A 2-lobed wheel will consist of four 
Fig. 32. sectors of 90° each, as EAHG, etc. 

A 3-lobed wheel will require six sectors of 60° each, etc. 
That these several wheels be interchangeable as derived from 
one spiral, Fig. 32, it is necessary that the sectoral arcs equal each 
other, while the sectoral angles differ. 

Thus, in Fig. 33, the arc DCE must equal the arc FCG, while 





Fig. 33. 

the angle DAE = 180°, and the angle BFG = 60°. 

In Fig. 32 the 180° sector DHG gives half the wheel A, Fig. 33„ 
while the sector EAJ, Fig. 32, gives a sixth of the wheel B, Fig. 
33, these sectors in Fig. 32 being selected by trial measurements on 
a drawing-board so that the arc DHG = arc EIJ. Likewise the 
sector FAI, Fig. 32, would make the eighth of a 4-lobed wheel, and 
HAE the fourth of a 2-lobed wheel, any two of which series of 
wheels from the 1-lobed to the 4-lobed and upward will work to- 
gether in pairs interchangeably. 

The velocity-ratio is always equal — ^. 

A ij 

2. As derived from two spirals of unequal obliquity. 

In Fig. 30 consider the wheel DEG as a 1-lobed wheel, to work 

with which a 2-lobed, 3-lobed, etc., wheel is desired. 

Here it is only necessary to proceed with, say, the steepest spiral 



LOGARITHMIC SPIRAL WHEELS. 



27 



first, just as was done in Fig. 32, obtaining sectors with arcs equal 
that of the corresponding steepest sector of the one-lobed wheel, 
and with angles a half, a third, etc., as great; resulting in the 
steepest sectors of the 2-lobed, 3-lobed, etc., wheels; then likewise 
with the flatter spiral, resulting in the flatter sectors, for the 
2-lobed, 3-lobed, etc., wheels; when the unequal sectors for a lobe 
of any one of the proposed wheels can be selected and the wheels 
laid out. 

Fig. 34 is an example of a 1-lobed wheel mating with a 3-lobed 




Fig. 34. 

one laid out in this way. Any two of the lobed wheels in the 
series above described will roll truly together. 

3. As determined from proportional sectors. 

In Fig. 35 let ADEH represent any given sector of a lobed 




Fig. 35. 

wheel. Eequired a sector AIJ of a wheel of three times as many 
lobes that will work correctly with the first-named wheel. In this 
case the sector AIJ must work correctly on the sector ADEH. 

Now for three times as many lobes in the new wheel, new sec- 
tors of the new wheel which mate with the first must have the 
sectoral angle one third that of the first wheel. That is, for the- 
sector AIJ to mate with sector ADEH, the angle I A J must equal 
a third of the angle DAH, and it is plain that the sector AIJ must 



28 



PKIKCIPLES OF MECHANISM. 



be similar to the sector ADE because of the equal obliquities. Also 
JN must equal HK, and IJ equal DEH. Therefore 
EL : HK : : AD : AI : : AE : AJ, 
a proportion which can be readily constructed graphically as in 
~K^ Fig. 36. With the mating sectors ADEH and 
AIJ thus found, the wheels may be laid out. 
If in the first wheel the other sector for a 
H lobe is not equal to the sector ADEH, but tha 
two for a lobe form an aliquot part of the full 
circle of 360 degrees, then a similar construction 
will give the new mating sectors for the other parts of the lobes of 
the new mating wheel. 

In this way the mating 4-lobed wheel is found for the given 





2-lobed wheel A of Fig. 38 as shown in Fig. 37, where EDF is the 
half of the given wheel. The number of lobes being doubled, the 
angles will be halved, and the halves of the sectors of Fig. 37 being 
similar to the new sectors required, are simply expanded. Thus 
the half AKD is expanded to the full new sector AHG, making 
HL = EJ. Similarly the half of ADF, or ADN is expanded to 
the full new sector AGI, and the new 
4-lobed wheel working with the former 
given 2-lobed wheel are both shown, 
mated, in Fig. 38. 

When one wheel is expanded to an 
indefinite number of lobes it becomes a 
straight notched bar, as in Fig. 39, of 
indefinite extent, the sectoral arcs of Fig. 38. 

which are straight inclined lines where A may be the driver, C the 
point of contact, and the center B at an infinite distance away. 
The wheel B thus becomes a sliding bar DE. The driver A may 
have unequal sectors when the notches will be unsymmetrical like 
the teeth of a rip-saw. 




LOGARITHMIC SPIRAL WHEELS. 

The velocity-ratio is the same as that for Fig. 27. 



29 




Fig. 39. 



EASEMENT CURVES. 

One characteristic of the above purely log-spiral wheels is the 
sharp abrupt intersection points, both salient and re-entrant, in tho 
outlines where the sectors lie 
adjacent. These if desired may 
be eased off into curves of any 
form provided they will roll 
properly, as in Fig. 40. 

Thus ADE is an easement 
sector mating with BHI, and 
AFG another mating with BJK. 
The easement sectors cannot be 




Fig. 40. 



log-spirals, and they may be constructed by assuming one, as, for 
instance, DE, and then the mating curve .ST must be found. How 
to thus construct mating curves is explained in a future chapter. 
To make such filleted outlines, DE and FG may be assumed, and 
the curves DF and GE drawn in as log-spiral sectors. Then as 
the position of the point B is not known exactly, an assumption 
may be made for it and tested by constructing a sector HIJK, 
which must be in proper relation to the full circle of 360 degrees. 

Second. Equal and Similar Ellipses. 

Fig. 41 presents a pair of equal and similar ellipses tangent to 
each other at C, with A, D, B, and E focal points. The ellipses 
are intentionally placed in contact so that the distance CH equals 
the distance CI, the points H and / being at the extremes of the 
major axes. If the ellipses be now rolled in mutual contact without 
slipping, in the direction such that H and /will approach, these 
points will meet when, at the same time, the major axes of the ellipses, 
will coincide with one and the same straight line through AB. 



30 PRINCIPLES OF MECHANISM. 

It is readily seen that Fig. 41 is symmetrical with reference to 
the common tangent CO, and that AC — EC, BC = DO. 

From a well-known property of the ellipse AG + CD equals 
the major axis of the ellipse, equals FH, a constant for all points 
of contact C. But as CD always equals CB, we have AC -f- CD 
= AC -f- CB = AB, equal a constant, and equals the major axis 
of the ellipse. Likewise for BGE. Hence if we take the focal 




Fig. 41. 

points A and B as axes of motion, the ellipses will roll truly in 
mutual contact as non-circular wheels. 

For the same reason that AB is constant, DE is also constant, 
and hence pins placed at D and E may be connected by a rod or 
link with freedom of movement about A and B. It is interesting 
to note that the elliptical wheels transmit the same motion from A 
to B as would a pair of cranks AD and BE and a connecting rod 
DE. 

The velocity-ratio for the elliptic wheels of Fig. 41 is 1 when 
the extremities of the minor axes are in contact at C, or when 
AC = BC; and least, or greatest in value, when the major axes 
touch at extremities on the line A B. 

If A is driver and revolves uniformly, then the fastest move- 
ment of B occurs when H and / are in contact, and slowest when 
F and G are in contact ; hence the ratio 





AH 






Fastest for B 


BI 


AH BG 


AH* 


Slowest for B ~ 


' AF 
GB 


~ BI'AF" 


BI* 



ELLIPTIC WHEELS. 



31 



This ratio increases rapidly with eccentricity. For example, in a 
nail- driving device for a certain boot and shoe nailing machine 
(see Fig. 130) with elliptic pitch lines, AH = 3.25 ins. and 
BI = 1 inch, so that the fastest for B is 10.56 times the slowest; 
and the absolute velocity of A being usually about 300 revolutions 
per minute, the slowest rate of absolute movement for B is there- 
fore r-^r- X 300 = 92.3 revolutions per minute, while the fastest is 

d./vO 



3.25 



X 300 



975 revolutions per minute. The ratio of these fig- 
ures is the same as by the formula givena bove, viz., 10.56. In this 
nail-driving device a crank is attached to B and by aid of a pitman 
or connecting link reciprocates a driving bar, the 
hastened motion of which bar resembled that of 
a driving bar accelerated by a spring, but possessed 
the great advantage of requiring no buffer to absorb 
the residual terminal shock, the crank and link 
always preventing the driving bar from going be- 
yond its predetermined lowest position.* 

Fig. 42 is an example from the Putnam Ma- 
chine Oo/s exhibit at the Centennial of 1876, of 
elliptic gears used on a slotting machine to ap- 
proximate the " quick return." The fastest mo- 
tion of the driven wheel is nearly 4.5 times the 
slowest. The distance between centers was 14 
inches. 

See also Fig. 133, for which the fastest for the driven wheel is 
41 times the slowest. 




Fig. 42. 



* A general expression for the velocity-ratio is obtained from the equation 
of the ellipse, viz.: 



AG 
BG-. 



n* 



— a. cos a 



m -f- a . cos /ff' 



from which we find the 



velocity-ratio = ^ = m + a ' C ° S fi f 



m — a . cos a 



where a and /3 are the angles DAG and IBG; and where 2m is the major di- 
ameter and 2n the minor diameter, a being \BE. 



32 



PRINCIPLES OF MECHANISM. 



The smoothness of outline of these elliptic wheels for rolling 
contact is favorable for their use in many cases where a variable 
velocity of the driven wheel is the main object, instead of some 
definite law of motion. 

As the curves here considered are only for pitch lines of gear 
wheels, it is plain that angles in those lines are by preference 
avoided. Compare Figs 42 and 45 with Figs. 71, 73, and 89. 

INTERCHANGEABLE MULTILOBED ELLIPTIC WHEELS. 

The remarkable elliptic wheels of the Eev. H. Holditch are 
the only ones that may properly come under this classification. 
These are called elliptic, not because made up of parts of ellipses, 
but because of the peculiar relation of this series of wheel curves 
to a corresponding series of ellipses as shown in Fig. 43. 

Draw the ellipse DPE with foci A and M and center 0. Draw 





' f 




S 










\ 




^U 










g/ / 


ex 

/ \ c 














/ <7 






i> 


JVi 








K H 


F LV 


A. 





°E 


G 


I 


L 



Fig. 43. 



OS perpendicular to AM, cutting the ellipse at P. Then for 
complete multilobed wheels, lay off OP in repetition on OS, giving 
points Q, R, S, etc., so that SB = RQ = QP — PO. Then 
through these points draw conf ocal ellipses as shown, that . is, with 
A and M as focal points. The points F, H, K, G, I, etc., may be 
found by making OF = OG = AQ, OR = 01= AR, etc., etc. 
Also we have Aa + aM — AQ -f QM, etc. The full elliptic arc 
may be drawn by stretching a thread AQMA around pins at A and 
M, and a pencil at Q. Then moving the pencil either way, kept 
tight against the thread, the ellipse is traced. 

The ellipse DPE is the unilobe as at A in Fig. 44. The bilobe 
is obtained from FQG, Fig. 43, by using the radii AF, Aa, Ah, Ac, 
etc., laid off on lines radiating from B, Fig. 44, at half the angles 
FAa, aAb, bAc, respectively, Fig. 43, this giving a sector CBT, 
Fig. 44, which is the fourth part of the bilobe required. 



ELLIPTIC WHEELS. 



33 



Similarly the trilobe is obtained by laying off the radii AH, 
Ad, Ae, etc., on lines radiating from B', Fig. 44, at a third of the 
angles HAd, dAe, etc., Fig. 43, thus giving a sector C'B'TJ, which 




Fig. 44. 



is the sixth part of the trilobe; from which the wheel can be con- 
structed, as shown in Fig. 44. 

In practice it is most convenient to make the angles equal to 
each other between lines radiating from A, Fig. 43; in which case 
the angles between the lines radiating from B, Fig. 44, are equal, 
and a half as great as in Fig. 43, and angles at B' ', Fig. 44, equal a 
third of those of Fig. 43, etc. 

The unilobe A, Figs. 43 and 44, is an ellipse as stated. But 
the bilobe is not an ellipse, though somewhat 
resembling it. Again, the wheel B' is far 
from being an ellipse, and the reason these 
wheels are called elliptic is because of the 
use of ellipses in Fig. 43 in constructing 
them. 

Fig. 45 is a photo-process copy of a pair 
of elliptic wheels like B and B' of Fig. 44, 
where the rolling curves serve as pitch lines 
for the teeth. 

The demonstration of the Holditch wheels 
is not simple, and will not be given here, 
but the wheels are admitted at this point 
because of their direct dependence upon 
ellipses. 

For the particular case that the number of lobes becomes infi- 
nite, the center of the wheel is removed to infinity, and what we 




Fig. 45. 



34 



PRINCIPLES OF MECHANISM. 



see of it is shown in Fig. 46, where the curve becomes a sinusoid 
LFHK, in which LH = nn, and GF = NK ' = Vm 2 - n> = AO, 
if m and n be taken as semi-axes of the ellipse. 




Fig. 46. 

To construct the ellipse on a drawing-board : The half length and 
breadth m and n having been determined, and laid off on OJ and 
01, make I A = ID = m. Then take any radius AP and describe 
an arc at P. Subtract this radius from 2 m, and with this differ- 
ence as a radius, equal DP, describe an intersecting arc at P. 
The intersection will be a point in the ellipse. Any number of 
points may be thus found and the ellipse drawn in. 

To construct the arc FH = HK, etc.: Take from a table of 
sines the values for every nine degrees up to ninety degrees, multi- 
ply these by Vm* — n*, and lay them off on ten equidistant perpen- 
diculars between H and G and including the latter, through which 

7tfl 

points draw the curve FH, GH being = — * 

The velocity-ratio is the same expression as that for Fig. 27, 
viz. : taking linear velocity of B as W, and angular velocity of A 
as V, we have AC. V= W, 



* Where the semi-axes of the ellipse are m and n the equation of the 
curve of the sinusoid LFHK is 



. x 
sin — = 

n y m a 



±y 



found by integrating the polar equation of the ellipse for dx = rdv and remem- 
.bering that r = m ±y. 



PARABOLIC WHEELS. 



35 



or 



W 
V 



Ac, 



making the rate of linear movement of B along the guides M and 
N the same as that of the end of a radius AG for the instant as 
revolving about the center A. 

Third. A Pair of Equal and Similar Parabolas. 

A parabola may be regarded as an ellipse so extended that one 
focus is at an infinite distance away. Thus in Fig. 41 suppose the 
focus B and D to be infinitely removed. Then F and G are ver- 
tices of parabolas, with A one axis of motion while the other, B, 
is at an infinite distance. In this case the motion of BG is simply 
to slide, as for B in Fig. 46. These rolling parabolas are shown in 




Fig. 47. 

Fig. 47, where A is the fixed axis of revolution for wheel A, while 
B slides in guides M and N. 

To prove that the parabolas will thus roll truly, we make use 
of Fig. 48 of two equal parabolas with their vertices in contact at 
G, and with A and focal points, A serving as an axis for A, and 
with B sliding in guides M and N. A property of the parabola 
is that the radial line AD makes the same angle as does D F with 
the tangent at D, or OF with the tangent at F. Also a well-known 
property is that the line AD equals the line DG equals FH, where 
OG and AH are perpendicular to AO. Now with A, M, and N 
fixed, suppose A to turn until D falls at K, at the same time sliding 
OB along. Then F will be brought exactly to K, with the latter a 



36 



PRINCIPLES OF MECHANISM. 



point of rolling contact on the line of centers A OK, as shown by 
the dotted lines, because AK = FH = GD, and the angles between 
the tangents to the parabolas at A^and the line ^4iTare equal to the 
angle between AD and the tangent at D, The same is true for all 




Fig. 48. 

the points of contact between the parabolas of Fig. 48, and hence 
the parabolas of Fig. 47 will roll truly upon each other, A turning 
about its pin while B slides in M and N. 

Fourth. A Pair of Equal and Similar Hyperbolas. 
In Fig. 49 let LCM and ICJ represent a pair of equal and simi- 

M 





Fig. 49. 



Fig. 50. 



lar hyperbolic sections for which A and E are focal points; the 
focus A being taken for the axis about which LCM rotates, and B 
the axis about which ICJ turns, the latter being mounted by a bar 



HYPERBOLIC WHEELS. 37 

BE, made fast to it, and centered by a pivot at B. A and B are 
then centers of motion, and BA C the line of centers, which must 
of course constantly pass through the point of contact G. 

To prove that these hyperbolas will roll truly, we refer to 
Fig. 50 of the same hyperbolas presented in correct relation with 
their asymptotes OP and OK, and focal points A and E. A well- 
known property of these curves is that the difference in length of 
any pair of lines AC and EG from any point C is the same for all 
points G on the curves, and is equal to the distance FG between 
the vertices and constant. Another property is that the tangent 
CH at C bisects the angle between the radial lines A C and EC. 

Now if we suppose the hyperbola A GF to be lifted from the 
paper as drawn, and be swung around till HG coincides with 
H' C , and the point C with the point C '; then the hyperbolas 
will be in common tangency at G f , and the distance from the new 
position of A over to B will just equal BG' — AG — FG, = AB 
of Fig. 49; and the two figures will otherwise agree: presenting 
conditions consistent with true rolling contact. 

The velocity-ratio is 

V _BC 

v ~ AC' 

In this case the point of contact is outside the space AB, and 
at any instant the wheels revolve in the same direction, while 
in former cases the contact is between the axes and the motions 
contrary. 

Fifth. Transformed Wheels. 

Let Fig. 51 represent any pair of correct rolling non-circular 
wheels; in this case sectoral. ^ p 

Divide the two wheels into mat- 
ing elementary sectors, for which a 
couple of mating short arcs may be 
referred to as a pair of arcs; a mat- 
ing couplet of radii as a pair of radii; 
and a mating set of angles as a pair 
of angles. 

Then the wheels A and B may be transformed in three ways, 
viz. : 

1st. By multiplying each of a pair of angles, or of arcs, by a 
constant. (Holditch.) 




38 



PRINCIPLES OF MECHANISM. 




Fig. 52. 



2d. By adding a constant length to all radii of one wheel, the 
arcs remaining constant. 

3d. By adding to one and subtracting from the other of a pair 
of radii a given quantity, arcs remaining unchanged. 

In these changes the elementary, or small, sectors are supposed 
to have angles of from three to ten degrees, according to desired 
accuracy of result, the precision of mathematical analysis not being 
expected, though a sufficiently close approximation to it is ob- 
tained to answer for most practical problems arising under this 
proposition. 

The second mode of transformation, applied to the wheel B of 
Fig. 51, by cutting off BB' from all radii, 
or from B to the circle DE, that is, adding 
negatively the length BB' to all radii of B, 
and bringing the cut-off ends of the re- 
maining portions of radii all to the point 
B' , will give the pair of wheels shown in 
Fig. 52 which will be found to answer the 
conditions of true rolling contact. 
The first mode of transformation applied to the wheels of Fig. 
52, by multiplying all the angles by two for a simple case, changes 
those wheels to the ones shown in Fig. 53. 
In this multiplying it is not necessary to use 
the same multiplier for all pairs of angles; 
indeed a different multiplier might be used 
for each pair of angles. 

The third mode of transformation ap- 
plied to Fig. 52, results in Fig. 54, where Aa 
is shortened and Be lengthened the same 
amount; Ab shortened and Bd lengthened the same amount; Ae 
lengthened and Bg shortened the same amount; and finally, Af 
shortened and Bh lengthened the same 
amount, these changes of the radii being 
made without any alteration of the lengths 
of the elementary arcs Ca, ab, Oc, cd, Ce, Cg> 
etc. It will be understood that the posi- 
tions of the radii will be changed, that is, 
that the angle CAa will be varied as Aa is 
shortened, while the length Ca is preserved ; 
and likewise for the other angles and arcs. 

These three modes of transformation are sufficient to change 





TRANSFORMED WHEELS. 39? 

any pair of correct rolling wheels into any other form. Thus, 
starting with the sectoral wheels of Fig. 51, suppose it were sought 
to derive a pair of complete non-circular wheels. 

Then, first, make the limiting radii of A equal, the one to the- 
other, also B, by the third mode; then by multiplying pairs of 
angles by a given quantity, one pair after another, as by the first- 
mode, we may make the total angle of one of the sectors 360 de- 
grees, giving a complete wheel, while the other will most likely not 
be complete. To make the latter complete also, apply the second 
mode of transformation until that wheel closes with 360 degrees. 

That the wheels thus rendered complete shall have the neces- 
sary shape as to eccentricity, law of varying velocity-ratio, etc., the 
wheels may be examined before entirely closed, and so treated by 
the modes of transformation as to bring about the desired law of 
velocity-ratio of the completed wheels. 

Extended changes of wheels in this way in general become 
tedious, and other processes on the drawing-board, hereafter ex- 
plained, will be preferred. The above modes of transformation 
will, however, be found most useful in making minor changes, such 
as rounding off a sharp prominence or depression, as in Fig. 40; 
and in certain special cases. 

INTERCHANGEABLE MULTILOBES BY TRANSFORMATION, 

The transformation of a pair of correct working wheels into- 
another pair, wider or narrower, 
by the first mode, is a compara- 
tively simple operation. 

Thus, in Fig. 55, we have a 
pair of Transformed Elliptic 
Wheels, mating half-ellipses A and 
B, in dotted lines, in which the 
angles are reduced by one half Fig. 55 - 

without changing the radii by which the 180-degree sectors are 
reduced to 90-degree sectors, or the one-fourth part of a pair 
of 2-lobel wheels, as explained in Willis, page 65. 

Other portions of rolling ellipses may be transformed into 
lobed wheels of intersecting characteristics, as explained in Mac- 
Cord's Mechanical Movements, page 54. 

As a second example, take the equal elliptic wheels A and 
B, Fig. 56, and transform A into a wheel of two lobes as at A'? 









L>\\ U 


\ A 




I 


*)) 1 I 






y / / / 
— * ii 



40 



PRINCIPLES OF MECHANISM. 




by adding a constant quantity to all the radii of wheel A, the 

elementary arcs remaining un- 
changed, as per the second mode 
of transformation. We will trans- 
form the half of A, the other half 
being like it except as to " rights and 
lefts." 

The amount to add to all the 
radii of A to obtain a 90-degree re- 
sulting sectoral wheel A' C M' to 
give exactly the fourth part of the 
wheel A' is not readily determined, 
several trial values being in some 
cases necessary. In the present case, 
however, we may refer to the Hol- 
ditch principle of Fig. 43, from 
which we find that by subtracting 
the distance, Fig. 56, AD from the distance AE we obtain EF 
for the amount to add to all the radii CA, GA, HA, etc. To 
apply this conveniently, draw the circle NRP to center A with 
radius EF. 

Then to construct A' CM' take A'C = GN, thus adding A J\ r 
to AC. Also make,4'£' = GQ, and C' G' = CG, thus determining 
the point G'. For the point H' make A' H' = HE, and G'H' = GH, 
thus determining the point //'. Likewise proceed to M' , making 
A'M' = PM. Trace the curve through the points C' G' H' . . . M\ 
completing the 90-degree sector A' C M' , with which the complete 
2-lobed wheel A' may be laid off. This wheel A' will work cor- 
rectly with the wheel B as a pair of 2-lobed and 1-lobed wheels, as 
A and B of Fig 44. 

The quarter wheel A' CM' is identical with the Holditch wheel 
found by the principles of Figs. 43 and 44, though the points 
G' ', H', etc., w r ill not coincide with the Holditch points even for the 
same number of points, and the same angle GAG, GAB, etc. 

To make a 3-lobed elliptic wheel by this mode of transformation 
make ES = ED, and subtract AD from AS. which difference use 
as the radius AT of a circle about A, from which measure the radii 
for the 3-lobed sector similarly as in the 2-lobed wheel. Combine 
these radii with the same arcs CG, GH, etc., for points in the 
60-degree sector for the sixth part of the 3-lobed wheel. 

Fig. 28 could be used as basis of a series of multilobed wheels 



TRANSFORMED WHEELS. 



41 




which would work interchangeably, spiral arcs and differences of 
limiting radii remaining unchanged. 

TRANSFORMED PARABOLIC WHEELS. 

As an example of a decided change in the appearance of awheel 
take those shown in Fig. 48, and h 

move the center of motion from A to 
A', Fig. 57, thus adding the distance 
A A' to all the radii, the elementary f|"|"| 
arcs being preserved in length. 

Divide the arc Cabd into equal 
parts and draw the radial lines aAi, 
bAj, etc. 

Then the radius of the point C of \~\ 
the transformed wheel will be A'C, 
and of the point e it is A'e, equal to 
aAi, where arc Ce = Ca; also radius 
A'f — bAj, and arc ef = ab, etc. 

The arc hCx of the transformed wheel, when CA' = \CA, will 
be nearly a straight line, but not exactly; because to be straight 
requires the contour of B to be a logarithmic curve resembling the 
parabola.* 

WHEELS OF COMBINED SECTORS. 

An interesting series of wheels may be obtained by combining 
sectors of the above various forms, as, for example, of the ellipse 
and log-spiral. 

In Fig. 58 we have for wheel A a 
90- degree elliptic sector A CD like the 
fourth part of B, Fig. 44, while the re- 
maining 270-degree sector is a log-spiral. 
Wheel B is a copy of A. 

In Fig. 59 we have a 2-lobed wheel A, 
each lobe of which is composed of a 
Fig. 58. 90-degree elliptic sector like BCT, Fig. 44, 

and a 90-degree log-spiral sector; working with a 3-lobed wheel B, 

* The equation of the logarithmic curve of B, making the mating curve 




JiGx a straight line, is y = a log 



2ax -\-x -\-a 



, where x and y are co- 



ordinates of the curve BC, and a == the distance A'C . The polar equation of 
the outline of the wheel A' is r = a sec v\ r being angle CA'e, etc. 



42 



PRINCIPLES OF MECHANISM. 



each lobe of which has approximately a 60-degree elliptic sector 
like B'C'TJ, Fig. 44, and approximately a 60-degree log-spiral sec- 
tor. If wheel A is first made with 90-degree sectors wheel B is 
best laid out by trial values of the sectoral angles, since the angles 





Fig. 59. 

will not be exactly 60 degrees in B. When a lobe of 120 degrees is 

found by trial for B, the remaining two lobes can be copied from it. 

In Fig. 60 DFC and CGE are a pair of rolling half-ellipses 

with axes of rotation at A and B» 
The other half of each wheel con- 
sists of log-spiral sectors ADH 
and AJC, separated by a circular 
sector AH J, laid out by first as- 
suming the circular sectoral angle 
CAI = HA J, then drawing in the 
log-spiral DHI, according to Figs. 20 and 21, then findingt he 
middle point H, and transferring the sector A HI to AJC, when 
the circle arc HJ completes the drawing of the half -wheel A. The 
mating half-wheel of B is a copy of ADHJC. 

In Fig. 60 the wheels, though composed of several sectors, are 
still similar, while in Fig. 61 we find sev- 
eral sectors combined in wheels that are 
quite dissimilar. Here CD and CG are a 
pair of elliptic sectors, DE, EF, aiid GH, 
HI log-spiral sectors, and CF and CI 
elliptic sectors. 

The above examples will serve to illus- 
trate the fact that a great variety of wheels 
may be made from the four curves named as the foundation. 

There are many cases, however, where none of the above wheels 
will answer, as, for instance, where a particular and definite laiv of 




Fig. 61. 



TRANSFORMED WHEELS. 43 

angular velocity foreign to the above curves must be closely fol- 
lowed during a part of, or even throughout, the entire revolution,, 
while the above wheels have laws of motion of their own which will 
rarely fit the special case of a predetermined law. 

The above wheels are fairly well adapted to the simple require- 
ment of a specified maximum or minimum velocity, or both, as occur- 
ring within a revolution, and at certain points therein. 



CHAPTER III. 
NON-CIRCULAR WHEELS IN GENERAL. 

In modern practical mechanism there exists a demand for non- 
circular gearing where the law of motion of the driven wheel is 
fixed by considerations foreign to log-spirals, ellipses, etc. ; which 
law the required wheels must follow, sometimes approximately, 
and sometimes exactly, in some cases for a part of, and in other 
cases throughout the entire revolution of the driven wheel. 

Again, there may be demanded similar wheels, differing from 
log-spirals and ellipses, and which can be readily drawn, and 
readily changed and redrawn to suit the designer's fancy. Hence 
there arise three cases, viz. : 

I. One wheel given, or assumed, to find its mate ; 
II. Laws of motion given to find the wheels / and 
III. Similar wheels. 

In either case the wheels may be complete or incomplete. 
Treating these separately, we have : 

I. GIVEN ONE WHEEL TO FIND ITS MATE. 

In Fig. 62, let A represent the assumed wheel so chosen that 

when the wheel B is found its cir- 
cumstances of motion will approx- 
imate those desired, more or less 
closely according to judgment in 
)B assuming A. When B is completed 
and tried, and found not to move as 
desired, A may be modified accord- 
ing to dictates of the first trial, and 
F 16 - 62 - B found again. 

In drawing the wheels the outline Cabc, etc., may be traced, and 
the center of motion, A, arbitrarily assumed. 

44 




NON-CIRCULAR WHEELS IN GENERAL. 45 

Then assume a point B for the center of motion for the wheel 
Cmno, and draw the line of centers AB. C is one point in the 
outline of B. Then describe an arc am, produced, with Cas the 
center. Also subtract Aa from the line of centers AB, and with 
the remainder, equal Bm, laid off from B, find the point m, as the 
second point in the wheel B. In other words, with Aa as a radius, 
and the center A, describe the arc as. With B as a centre and Bs 
as radius, describe the arc sm, when the intersection m is the second 
point in B. Next, with ab as radius and m as centre, describe the 
indefinite arc at n; and with Ab as radius, describe the arc bt, and 
with Bt as radius, describe the arc tn, to find the intersection n,. 
as a third point in the w T heel B. 

Thus we have Cm = Ca, mn = ab, no = be, etc. ; and Bm = 
AB - Aa, Bn = AB - Ab, etc. 

Proceed thus for the entire circumference of A, resulting in 
corresponding or mating points for B. A curve traced through 
these points gives the first trial outline for the wheel B. 

Now if, when the center point B is assumed, it makes the line 
of centers AB too great, the wheel B will have a gap in it; and con- 
versely if AB be taken too small there will be an overlap. 

( To make B complete, that is, to just close up without a gap or 
overlap, a new point for B must be assumed, and the whole work 
of finding the points mno, etc., repeated, one or several times, until 
sufficient closeness is obtained, and the outline of B traced, thus 
completing the pitch lines, or wheels, A and B. 

The distance to move B will be not far from one-sixth of the 
arc of the gap, or overlap. 

The above tentative process, as unscientific as it may appear, 
is nevertheless the only way for the case of one wheel given to 
find its mate. 

As to accuracy, to make the spaces Cabc, etc., excessively great 
introduces an error due to the difference between the chord and 
arc ; while to make them excessively small, the errors of graphical 
work accumulate. Probably 10-degree angles around A will be 
found about right. 

To increase the length of the steps Ca, ab, etc., without jeopard- 
izing accuracy, Eankine makes the ingenious application of the 
following 

Graphic Rules for Equivalence of Lines and Arcs. 
First. To find a straight line equal to a circle arc. 



46 



PKItfCIPLES OF MECHANISM. 





To find the tangent line DF = the arc DE, Fig. 63, take 

f/ DO = \DE, and with as a center, describe 

the circle arc EF, cutting DF in F. Then 

DF = arc DE within less than a tenth of one 

per cent, up to an angle of 30 degrees, and in- 

E D "o creases as the fourth power of the angle. 

Fig. 63. Second. To find a circle arc equal to a 

straight line. 

To find the arc DE = line DF, Fig. 64, take DO = \DF, and 
draw the circle arc FE from as a center. 
Then arc DE = line DF within same limits 
as first rule above. 

The arc DE must be tangent to the line 
DF at D. Hence the center of the arc DE 
is on DG drawn perpendicular to DF at D. 
The same rule and construction apply for all 
arcs starting from D, with centers on line DG, 
as for the dotted arcs shown. 

The above percentage of error is true for circle arcs only; other 
curves give rise to greater errors. 

To Draw a Tangent to any Curve. 

To draw a tangent line to the curve DOE at the point 0. 
First draw any curve FG, cutting the curve DOE, and also a 
series of straight lines as DI, He, ac, etc. 
Then lay off the length DF from to 
2, noting the point /; Oe from G to H, 
V, noting the point H\ ao from o to c, not- 
ing the point c, etc. Then trace a curve 
through the points Hcdl, etc., as shown, 
noting its intersection T with the curve 
Fig. 65. j?@ m Then a straight line through the 

joints T and will be tangent to the curve DOE at 0. 

These rules will be found useful in the graphic operations of 
problems in Mechanism. 

Application of Rule Second to the problem above, of one wheel 
given to find its mate. Here ACDE, Fig. 66, is a portion of wheel 
A, and B the proposed center of wheel B, with AB the line of 
centers. 

Draw Cd tangent to the wheel A at the point C. With a as a 
center, taking Ca = \Cd, draw the circle arc DdF extended. Draw 




NON-CIRCULAR WHEELS IN GENERAL. 



47 



ihe circle Dm from A, and the circle mF ivom. B, giving the inter- 
section F of arc mF with arc DdF. Then F is a point in the per- 
iphery of B, since arc CD = line Cd = arc CF, of B, according to 
rule 2d above. 

To find another point in B, draw the tangent De at D, and the 
radius AD extended, noting the angle gh. Then draw the radius 
BF, and the line Ff at the angle with FB of ij — gh. Take the 
points c and b at the J distance Z?e from i> and from F, Then 




Fig. 66. 



with be as a radius and center b, draw the arc eF; also from center 
c the arc /£ extended. Then, with a radius AB — ^4i? and 
center at B, cut the arc fG at 6^. This latter point is another 
point in the contour of wheel B, since arc DE = line De = line 
Ff = arc FG of £. 

Thus proceed for the whole circumference of A; and if the cir- 
cumference of B closes, as by center B having been rightly chosen 
in distance from A, the points C,F, G, etc., will be correct points in 
the contour of B. But if wheel B does not close, assume another 
position for center point B and repeat the work for a new set of 
points C,F,G, etc. Finally when the wheel B closes, the points 
C,F,G, etc., will be correct points in the arc of B, though they will 
be too few to admit of drawing in the contour of B. 

A sufficient number of points intermediate may then be found 
as by the method of Fig. 62, thus completing the wheel B. 

The advantage of the wide steps of Fig. 66 is to permit of find- 
ing the correct position of the center B with comparatively little 
work. 

In Fig. 67 is a photo-process copy of a pair of wheels in metal, 



48 



PRINCIPLES OF MECHANISM. 



the pitch line of the lower one of which was first traced by pencil 
free-hand, and the center point assumed, when the upper or mating- 
wheel was found by the process above described. The wheels ar& 
about 6 inches between centers. 





Fig. 67. Fig. 68. 

Fig. 68 is a drawing copied from a pair of wheels used on a 
paper-cutting machine from England and exhibited at the World's 
Fair of '93. The wheels were in cast iron, about 16 inches between 
centers, in which the smaller is simply an assumed circle, with its 
axis chosen arbitrarily in eccentricity, while the larger wheel is 2- 
lobed. 



II. LAWS OF MOTION GIVEN TO FIND THE WHEELS. 

The centers of a pair of wheels chosen to illustrate this case are 
taken at A and B, Fig. 69, with AB the line of centers, and the 
heavy outlines CDFH. . .X, and CEGI . . . Y, the contour lines of 
the completed half-wheels, or at least as much as lies above the 
extended line of centers. 

The wheel A in Fig. 69 is supposed to revolve about its center 
of motion with velocity constant, so that the "law of motion " for 
A is very simple, viz., motion uniform. A semi-circle aceg, etc., is 
drawn within the half-periphery of A and divided into six equal 
angles or sectors each of 30 degrees. For convenience suppose the 
wheel A to make a half -revolution in six seconds, or any one of the 
six sectors in one second. Then the law of motion of A is repre- 
sented graphically by the sectors into which it is divided, these 
several sectors passing over the line of centers in equal times, say 



NON-CIRCULAR WHEELS IN GENERAL. 



49 



one second for each. These angles and sectors may be called 
velocity angles or sectors, or sometimes auxiliary angles or sectors. 




Fig. 69. 

If for B the sectors were equal also, its motion would be uni- 
form, and the resulting wheels would be circular and without 
special interest. But let the sectors of B be varied in angle as 
shown, any one sector passing the line of centers per second. Then 
the law of varied motion of B may be represented graphically by 




the varied sectors, as shown. Hence the laws of motion of the 
wheels are represented by the angular spaces laid off around the 
centers A and B. Though A, here, has equal angles and uniform 
motion, it may readily have a varied motion and a correspondingly 
varied series of sectors, as for the case that the driver A has a 



50 PRINCIPLES OF MECHANISM. 

varied motion. But in practice the driver will usually have uni- 
form motion. 

Next, suppose all the sectoral angles to have sectoral arcs drawn 
in, these arcs being all of the same length, so that the narrower the 
sector the greater will be its length. Thus the angles for A being 
all the same, the arcs are all the same distance from A in a circle 
arc. But for B the arcs arrange themselves at varied distances 
from B, as shown in Fig. 69. 

The sectors of A and B may be regarded as in pairs, so that if 
ac should roll against hj the sectors would turn through their angles 
in the same time, and the velocity-ratio would be the inverse ratio 
of the sectoral radii. The same being true of the next pair of sec- 
tors rolling upon each other, it follows that the velocity-ratio 
would change from the one pair of sectors to the next, in case the 
radii differ, as in fact they do. 

Thus for A moving uniformly, B would have a varied motion 
according to its sectoral angles, or according to the stated law of 
motion, since the angles were laid out according to that law. 

Hence if these wheels, composed of stepped sectors, could be 
kept revolving with these sectoral arcs in rolling contact one after 
another in order, the law of motion, in general, would be correct. 
But this would be thoroughly impracticable, since it would require 
the distance between the centers A and B to be continually vary- 
ing. This is obviated by changing the radial lengths of the sectors 
so that their ratio in pairs will be the same as before, and their sum 
equal the line of centers AB, while at the same time the modified 
sectoral arcs are merged into each other in continuous lines, as 
shown in the lines CDFH. . . Xand CEGI. . . Y. 

Fi°\ 70 will serve best to show how this may be done, where A and 
B are the centers, cAe a sector for wheel A, and kBm a sector for 
wheel B, where are cde = arc him, these sectors with lettering and 
construction lines being taken from Fig. 69. Now to find points 
F and G in the final perimetric arcs of wheels A and B, such that 
the velocity-ratio for F and G in driving contact shall be the 
same as that for the arc cde rolling on the arc Mm; it is only nec- 
essary that 

length AF _ Ad 
length BG~ BV 

By graphic process this is most readily done by drawing lines 
from A and B to meet in some point 0, and parallels thereto from 



NON-CIRCULAR WHEELS IN GENERAL. 51 

if and L meeting in u, where cM is an arc struck from A, and JcL 
an arc struck from B, and a line OuN, when 

AM = Ad : AN :: BL = Bl : BN. 

Now with radius AN find the intersection F by drawing the arc 
NF; likewise with the radius i?JVfmd the intersection G; dF be- 
ing drawn from the center point A and the middle of ce, and Gl 
likewise drawn with respect km and B. 

Fig. 70 is taken out of Fig. 69, with like lettering; and by re- 
ferring to Fig. 69, we find that other points, as DHJ, etc., and 
FJIK, etc., were obtained in the same way. Through these points, 
when all are thus determined, the contours of the completed half- 
wheels A and B may be drawn in with smooth outlines. 

In practice the angles of the sectors should not exceed 10 to 
15 degrees for suitable accuarcy of results. 

In the figure we have drawn only half of each wheel A and B, 
but of course the remaining halves may be drawn in the same 
way. 

In practice these wheels have been made where the specific law 
of motion of B was carried in some cases through 180 degrees, and 
in others through the whole 360 degrees. At the Centennial of 
1876 there was exhibited, by B. D. Whitney of Winchendon, 
Mass., a certain barrel-stave sawing machine where the carriage 
carrying the blank block for staves was fed against the " tub-saw " 
.at a uniform rate of advance and with a quick return; the move- 
ment of the carriage being made by a crank and pitman connec- 
tion driven by non-circular gears of Fig. 71, 15^ inches in diame- 
ter. On examining the marks of the saw on sawn staves it was 
found that the forward feed was wonderfully close to a uniform 
rate of advance, proving that the wheels must have been care- 
fully worked out as to laws of motion, even allowing for a short 
pitman. 

In this case the wheel A made something like a three-fourths 
turn, while B made a half -turn with uniform motion of stave block. 

Fig. 71 is an accurate copy, the wheels serving as type, which, 
with printer's ink and paper, gave the copy that is photographed 
ior the figure. 

The divergence of the teeth near the salient angles is noticeable, 



52 



PRINCIPLES OF MECHANISM. 



and suggests blocking of the gears, and yet they actually operated in 
the most satisfactory manner. 

Also see Fig. 120 for another example of careful laying out. 

In another example of these wheels the pitman was in effect of 
infinite length, giving uniform motion to a carriage forward, with 




Fig. 71. 



a quick return, the driver revolving uniformly for about a three- 
fourths turn, to the half turn of B, as required in a boot and shoe 
screw nailing machine. See Figs. 134, 135, and 136. 

As a case of quick return with short pitman with a little more 
latitude for return, and for starting and stopping at the ends of the 
uniform motion, see Fig. 72. 




Fig. 72. 

At the Centennial the problem was made known of a motion for 
winding yarn on a conic bobbin, such that the yarn would be taken 
off a reel uniformly when winding upon the cone from a base of 
four inches to a tip of one inch, back and fourth on the cone in 
regular screw pitch. This requires the cone to revolve faster when 
winding yarn on its tips than when winding on its base, and vice 
versa. 

This problem has since been worked out in unilobed non-circu- 




NON-CIRCULAR WHEELS IN GENERAL. 53 

lar gears, such that while the driver A revolves uniformly, the 
follower B will have an acceleration for a 
half-revolution during the windings down 
the cone, and then a retardation for a half- 
revolution of B during the winding up the 
oone. Thus one revolution of B takes place 
while winding down and back the cone. Now 
if the cone is to have several convolutions of 
yarn, say ten, in winding down, and then ten in 
winding back, the cone must make twenty rev- 
olutions to one of B. This relation can be 
obtained by interposing circular gearing be- 
tween the cone and B, with a ratio of in- 
crease of speed of twenty to one. Fig. 73. 

These non-circular gears, as unilobed wheels in metal, are shown 
in Figs. 73 by photo-process copy. 

In Fig. 89 these same wheels are constructed as bevel non-cir- 
cular wheels. 

In Figs. 76 to 78 the question of determination of the pitch 
lines of the non-circular wheels for this bobbin-winding problem is 
fully solved as a particular example in illustration of the applica- 
tion of the principles of Figs. 69 and 70; the example answering 
to the case of law of motion definite and unalterable for the entire 
revolution. 

Even this is simpler than the most general possible case where 
the law of motion for the second half of the revolution of the 
driven wheel is different from that of the first half, as it may be; 
for which case the resulting wheels are non-symmetrical, and may 
resemble the wheels of Fig. 67, for which wheels, of course, assum- 
ing the driving wheel in uniform motion, the driven wheel follows 
some law of motion, simple or complex, which law, how T ever com- 
plex, may be made out and expressed graphically, where, instead of 
using velocity sectors as in Figs. 74 and 78, passing time may 
represent the abscissas, and where the ordinates represent veloci- 
ties of the driven wheel for the corresponding moments of time or 
corresponding points of revolution of the driver. 

As an example of slightly unsymmetrical wheels, suppose the 
tool of Fig. 75 were required to return at a uniform rate of 
motion, as well as to advance according to that law. That return 
may be in the same time as the advance, for which case, if B is in 
the prolongation of the line SU, the wheels would be symmetrical. 



54 PRINCIPLES OF MECHANISM. 

Bat they would not be symmetrical when taking B above SU as 
shown; and still more unsymmetrical if the return of the tool, still 
uniform in velocity, were to be in less or more time than the ad- 
vance. For this, according to the description of Figs. 74 and 75, 
the scheme of velocity angles of the lower half of B is to be made 
in the same way as the upper half shown. Then A is to be divided 
into two parts, apportioned according to the assignments of time 
for the advance and the return of the tool T, those parts to be di- 
vided into velocity sectors corresponding in numbers with those of 
the mating parts of B. 

SOLUTIONS OF SOME PRACTICAL PROBLEMS. 
FOR LAWS OF MOVEMENT GIVEN TO FIND THE WHEELS. 

As this subject is evidently one of considerable importance in 
practical mechanism, the solution of several problems is here given 
to illustrate the application of the above principles. 

1st. The Shaping-Machine Problem. — This is called the shaping- 
machine problem because on the machinist's shaping machine the 
cutting tool is desired to have a uniform motion forward for the 
cutting stroke, and a quick return stroke where the tool returns 
idle and should do so quickly to save time. 

For this we have given a shaft, A, revolving uniformly, gear- 
ing with a second shaft, B, revolving at such variable rate of rota- 
tion as to secure the uniform forward and quick back stroke as 
described; also a carriage for the tool connected to the driven 
wheel B by pitman and crank. 

In Fig. 74, T is the tool to move along the cut QB, held on the 
slide ME and guided by NO. The slide EM, carrying the tool, is 
connected to the crank BD by the pitman DE, E being the " cross- 
head " pin moving in the line or path ST7, which pin moves with 
the tool. 

Divide the path ££7 into equal parts to represent uniform mo- 
tion, the pin passing over these parts forward in equal times. The 
pitman DE = FS, = HU, = GV, etc., as shown. Hence the 
crank pin D must move through the arc, spaces EG, GD, DI, IJ T 
JK, KH, in equal times, from which we have the angles FBG, GBD, 
DBI, etc., to represent the law of motion of the driven wheel B for 
the forward stroke. The back stroke being made quick, simply, 
we may assume the angles HBL, DBP, and PBF, which for the 
quick return should be larger than those for the forward stroke. 

Thus, by aid of Fig. 74, we obtain the angles to be passed over 



SOLUTIONS OF SOME PRACTICAL PROBLEMS. 



55 



by the wheel B for its entire revolution. Taking WXior the line 
of centers through A and B, we may transfer the velocity angles to 
Fig. 75 with like letters. 




Fig. 74. 




In Fig. 75, as was done in Fig. 69, draw equal arcs in the 5 
angles about B, as ac — de — fg, etc., and extend radii from their 
center points as shown, nine in all. Taking A to revolve with 
uniform velocity, there will be nine equal angles and arcs about 
the axis A as shown, these arcs being equal those of B, viz. : 
mn = no, etc., = ac = de, etc. Draw extended radii from the 
centers of these arcs as shown. Then the final radii for the fin- 
ished wheels may be found by graphic proportion, by aid of the 
triangle ABO, as explained in Fig. 70. Thus make Bh = Bo, 
lis parallel to BO, us parallel to AO, and draw the line Ost. 
Then make Ar — At, and Bp = Bt, and we have two points r and 



56 



PRINCIPLES OF MECHANISM. 



p in the final peripheries of the wheels A and B. The other points 
are found in a similar manner, as explained in Fig. 69, and the 
final wheels are drawn in by tracing lines through the points thus 
found. 

The curves thus found, it is to be understood, are to serve as 
the pitch lines of a pair of gear wheels. Figs. 71 and 72 are ex- 
amples of completed wheels by this process. 

The shaper in practice will usually have the crank pin adjust- 
able to different radial distances. In this case the above solution 
should be made for the crank at some mean position, where it is 
likely to be most used, as the law will be slightly deviated from for 
a different position of the pin. 

2d. A Bobbin Winding Problem. — This problem, stated above, 
of winding yarn on a conical bobbin, or " cop," while feeding on 
the yarn spirally up and down the cone, and varying the speed of 
revolution of cone so as to draw the yarn at a uniform rate from 
a reel to avoid varied tension of yarn while winding, is believed 
to possess sufficient interest as an application of Fig. 69 to warrant 
its practical solution here. 

Pig. 76 represents the conical bobbin with a spindle-like exten- 
tension, showing also the cop of yarn in section. 
This is said to be the best form of bobbin and 
cop from which to supply yarn to knitting 
machines, since the yarn will lift off, and up- 
ward, from the position of Fig. 76, with the 
slightest and with the most uniform resistance — 
conditions of utmost importance to securing a 
smooth knitted web. 

To preserve the same cone form of cop from 
end to end, the yarn must be fed upon the cone 
in a spiral or screw of uniform pitch, from base to 
spindle at vertex, and vice versa. The number of 
convolutions winding up, and the number back 
again, is arbitrary, and perhaps should be 20 to 40. But to simplify 
the present illustrative example, take it four ; that is, four turns of 
the bobbin in winding from base to spindle, and four turns in wind- 
ing back, eight turns in all. This multiplying of the motion, 8 to 1 , 
may be done by using circular gearing between the non-circular 
gear wheel B and the bobbin, causing the bobbin to turn eight 
times as fast as B. This gearing, as by a screw of uniform pitch, 




Fig. 76. 



SOLUTIONS OF SOME PRACTICAL PROBLEMS. 



57 



is to work the feeding eye through which the yarn is to pass in being 
guided upon the cone. 

Developing the cone with a single layer of thread upon it, we have 
the sector ODE, Fig. 77. DG rep- 
resents one convolution of yarn, KN 
another, LP another, and 1J the last. 
These are Archimedean arcs, and may ^ 
he put in a continuous spiral by shift- 
ing the second convolution KN around 
the center till KN falls at GH, LP 
at HI, meeting IJ in the case the cone 
is so assumed, as here, that DE is the 
third of the circle. The whole thread 
in winding down the cone is then 
DGHIJ, and in winding back is the 
same taken in the inverse order. 

Now divide this thread into a certain number of equal parts, 
DQ, QR, RS, ST, TU, UJ, six in all. These portions must be 
w T ound upon the bobbin in equal times, in order to take the thread 
uniformly from the reel as required. Therefore the angles DOQ, 





Fig. 78. 



QOR, EOS, etc., must be passed in equal times by the finished 
wheel B, while the wheel A moves through equal angles in the same 
equal times. 

Tn Fig. 78 lay off the six angles hBi,jBk, IBm, etc., as shown, 
varying in the same proportion as do the angles DOQ, QOR, ROS, 
etc., in Fig. 77; taking care that the six angles in Fig. 78, added 



58 



PRINCIPLES OF MECHANISM. 



together, equal 180°. For this result we have the added angles in 
Fig. 77 equal 360° + 120° = 480°; and hence multiply the angle 
DOQ by iff = }, giving the angle hBi, Fig. 78. Also multiply 
angle QOR, Fig. 77, by f, giving the angle jBTc, etc. 

Then draw in circle arcs hi, jk, Im, etc., equaling correspond- 
ing arcs in A, and proceed as in Fig. 73 as shown, completing the 
half-wheels A and B, for which all the work is given in Fig. 78. 
The other halves are the same, A being symmetrical with respect to 
the line AC as an axis of symmetry, and likewise for B. 

These wheels are one-lobed; A, the driver, differing somewhat 
from B in shape; and as constructed carefully in metal by above 
process, are shown in Fig. 73 for the same problem. 

Motion of Driver A, a Variable.~In all the above problems, the 
velocity of the driver A has been supposed to be constant, but the 
solution is readily made where its velocity is variable, for which 
case it is only necessary to make the series of velocity sectors un- 
equal to suit the varied motion, as was done for B, instead of 
making them equal. 

Case of Multilobed Wheels. — In the above wheels, both driver 
and driven, have been treated as unilobed, but multilobes are pos- 
sible, particularly for the driver. For this, for a bilobed wheel A, 
it is only necessary to put into the scheme of angles for A twice as 
many velocity-angles as for B, three times as many for a trilobed, 
etc. ; and similarly for a multilobed wheel B. 

III. SIMILAR WHEELS. 

1st. Case of Similar and Equal Unilobed Wheels from As- 
sumed Auxiliary Angles. — In Fig. 79, A and B are centers of mo- 




tion of the wheels. About A draw a series of auxiliary sectors, and 
about B the same series of sectors. Then, according to Fig. 69, 
draw equal arcs in all the sectors, and extended radii from the 
centers of the arcs as shown. Then the proportional radii, as of 



SIMILAR AND EQUAL WHEELS. 59? 

the auxiliary sectors and of the perimeters, may be found, when 
the wheels may be drawn in according to Fig. TO. 

Fig. 79 shows the halves of a pair of wheels. These may be 
copied for the opposite halves, when the wheels will each be sym- 
metrical; or another set of different auxiliary sectors may be drawn 
below AB and unsymmetrical similar wheels produced. 

Fig. 129 is an example of a pair of wheels of this kind in metal. 

2d. Similar and Equal Multiloled Wheels. — In Fig. 80 we have 
the half of a 3-lobed wheel where the lobes are all similar to 
each other, and one wheel similar and equal to the other, but where 
the lobes are unsymmetrical. 




Fig. 80. 

The wheels are arrived at by assuming a system of auxiliary sec- 
tors in the 60° angle CAD, and a like system in the 60° angle 
CBG. Determining the proportional radii between sectors and 
perimetric points as before, and tracing in the outline curves DC 
and HC, we find them similar. 

Again, assuming a new set of auxiliary sectors, one like the- 
other, in the 60° angles DAK and FBH, determining proportional 
radii, and drawing in the perimetric curves DK and FH, we will 
find completed a third of the wheel A ; also a third of the wheel B. 

By repeating these perimetric curves around through the remain- 
ing 60° angles of the wheels, we complete the pair of equal and 
similar 3-lobed wheels. 

Thus equal and similar wheels of any number of lobes can be 
drawn. 

3d. Dissimilar Multiloled Wheels. — Each wheel of Fig. 80 has 
lobes that are similar to each other, but they are evidently not 
necessarily made so, because each pair of divisional angles CAD and 
CBH&re here equal to each other, but not necessarily equal to the- 
next pair of divisional angles; and besides, the systems of aux- 



■60 



PRINCIPLES OF MECHANISM. 



iliary angles and sectors in one pair of divisional angles are not 
required to be the same as in any other pair. 

4th. Multilobes of Unequal Numbers of Lobes. — In Fig. 81 
we have a 3-lobed wheel mating with a 2-lobed one, where the 




Fig. 81. 

ialf-lobe CAD of A mates with the half-lobe CBE of B; and 
the half -lobe CAF with CBG. The sector DAFC is the third 
part of A, and EBGC the half of B. 

In a similar manner, A may have any assumed number of lobes, 
and likewise for B. 

In Fig. 81 the angle of the half-lobe CAD = CAF, and also 
CBE = CBG, but they may be unequal, though it would seem ad- 
visable for good results that the half-lobe angles have the relation 



CAF 
CAD 



CBG 
CBE' 



and that adjacent pairs of auxiliary angles in adjacent half-lobes be 
not greatly different in ratio of radii, that the perimetric curves 
may enter upon each other at the limits of sectors. 

But it is plain that the auxiliary angles in any half-lobe are 



DISSIMILAR WHEELS. 61 

entirely arbitrary except at the lobe limits as above stated. In this 
way the auxiliary angles of a half-lobe may be changed at pleasure, 
thus modifying the perimetric arcs to suit fancy or requirements; 
it being only needful to note that the half-lobes that come to mate 
together be related in principle, as for the case where the lobes in. 
one wheel are even and in the other odd. 



CHAPTER IV. 
Case II. Axes Meeting. 



SPECIAL BEVEL NON-CIRCULAR WHEELS. 

It is usually advisable to refer wheels of this kind to spherical 
surfaces which are normal to all the elements of contact of the 
rolling cones whose vertices are at the center of the sphere and 
hence called normal spheres) the results of practical problems in 
drawing these wheels being either curves upon actual normal 
spheres, or ordinates, angles, etc., by which the curves may be traced 
upon the sphere by the pattern-maker as he follows the draftsman 
in the construction of these wheels. 

Following the same order here as in plane wheels, we have — 

First. The Equiangular Spiral. 

1st. One-Lobed Wheels.— Take Fig. 82 to represent two views 
of the normal sphere on which a pair of spheri- 
cal equiangular spirals are drawn for a pair 
of wheels A and B. The outline of one wheel, 
A, is shown at Cabc . . ./ and C'a'o'c' . . ./' in 
the two projections; CBh, C'B' being the 
mate; a copy of the first one, A Cf. 

To draw this wheel, the actual normal sphere 
is to serve best as the drawing-board. From 
A A' draw a series of meridians at equal inter- 
vening angles of not far from 5° each. In 
Fig. 82 they are at 30° to avoid confusion of 
figure. We may assume a point C, C ', making 
A C the shortest spherical radius of the proposed 
wheel. Then draw the line Ca, Ca', making a 

certain angle with the meridian lying midway between AC and Aa. 

From the point a, a' where Ca intersects the meridian A a, draw a 

62 




Fig. 



AXES MEETING. 



63 



line ab, a'V ', making the same angle, on the surface of the sphere, 
with the meridian midway between Aa and Ab that the line Ca 
did with the meridian between A C and Aa. This line intersects 
the meridian Ab at b. From b draw the line be, b'c' , making the 
same angle with the meridian midway between Ab and A C as the 
previous lines did with their meridians. 

So continue till the point/,/' is reached, giving a half- wheel of 
one lobe. The points a'b'c', etc., may be symmetrically copied upon 
the other side of C'A'f, thus completing the drawing of a one-lobed 
equiangular bevel non-circular wheel. This wheel may be copied 
at CBk for a mate, B, when we have the drawing of a pair of bevel 
non-circular wheels. 

These drawings may, however, have been made upon normal 
spheres in the form of two separate spherical segmental shells in 
wood of uniform thickness, as shown in Fig. 83. With drawing 
completed, the pattern-maker could cut the 
wood to the finished wheels shown, following 
the drawing thus laid out for him. The draw- 
ing, however, as above described, is usually for 
the " pitch line" of a gear wheel, and the teeth 
may be laid out before going to the pattern- 
maker. 

Where the mating wheels, as in this case, 
are both alike in pitch line, and perhaps for 
teeth as well, it is only necessary to complete 
one drawing and pattern to obtain castings 
for any number of pairs of cast wheels. 

One point not to be overlooked is in regard 
to the size of the wheels as above explained. They may come out 
so as to require less or more than 90° between the axes, and for 
them to come out with any previously assumed angle of intersection 
of axes, several trials of tracing the line Cabc . . ./ may be required. 
Any one of these trial curves, however, will work correctly with 
another copied from itself. The angle between the axes is the 
same angle as COf, which can be measured as soon as the curve 
Ccf is traced. 

This curve of Fig. 82 differs from that of Fig. 95 of Mechanical 
Movements, of Professor 0. W. MacCord, for the reason that there 
the lengths of the meridian-arc radii are made equal the radii of 
the plane equiangular spiral, which process makes the spherical 
spiral arc cut the meridians at a sharper angle for long than for 




64 PRINCIPLES OF MECHANISM. 

short radii. This is most easily seen to be true in a quite eccentric 
1-lobed wheel, in which, for short radii, the steepness of the 
curve on the sphere is nearly the same as for the plane spiral, while 
for the longer radii, nearly 90° of meridian arc, the distances from 
point to point are much less on the spherical than on the plane 
spiral, while the rate of change of meridian-arc radii remains the 
same as for the plane spiral. It would appear from this that the 
proposed wheels of Professor MacCord cannot work, because for 
both wheels of a pair the obliquity near the axis A or B differs 
from that for points more remote; while, as the wheels engage in 
action, the shorter radii of one wheel mate with the longer of the 
other where the obliquity is different. 

2d. Two-Lobed Wheels, Multilobed, etc. — Having the curves of 
Fig. 82 drawn on the actual normal sphere, we may take the sectoral 
part C A'c' as the quarter of a two-lobed bevel non-circular wheel, 
a pair of which will work correctly. Or again, the sectoral part 
c'A'f is the quarter of a larger wheel, etc. With a spiral thus 
drawn on a sphere a wheel of any number of lobes can be made 
out, any one of which wheels will work with another like it. 

That the axes of these wheels be at right angles, the wheels 
must be of certain size and may be made out by trial measurements 
on the spherical drawing. This tentative process, though seem- 
ingly tedious, is yet probably less troublesome than mathematical 
analysis, which in problems of mechanism often become tiresome. 

3d. Interchangeable Multilobed Bevel Non-Circular Wheels. — 
Having drawn a spherical equiangular spiral as in Fig. 82, a sector 
of 180° may be selected from it for the half of a 1-lobed wheel. 
A second sector may be selected, of 90°, for the quarter of a 
2-lobed wheel whose arc equals that of the 180° sector. Again, 
a third sector may be selected, of 60°, for the sixth part of a- 
3-lobed wheel and whose arc equals that of the 180° sector. 

Any two of these wheels will work correctly together. 

The sphere will put a final practical limit to the number in the 
series of these wheels, for a given spiral. Another spiral will fur- 
nish another series of interchangeable wheels. 

Pairs of the above wheels selected to work together will have 
various angles between the axes, the point of intersection always 
corresponding with the center O of the sphere of Fig. 82. 

Again sectors of different spirals may be combined in a lobe 
making unsymmetrical lobes. 



axes meeting. 65 

Second. Elliptic Bevel Non-Circular Wheels. 

1st. One-lobed Wheels. — This case is admirably treated by Mac- 
Cord in Mechanical Movements, Fig. 88, where spherical ellipses are- 
shown as traced upon the normal sphere by means of two pins 
fixed at the focal points, around which is placed a loop of inelastic 
thread, when a pencil, drawing the thread tight and moved along: 
against the tense thread, traces the "spherical ellipse" in a manner 
entirely similar to that so well known, of tracing the ellipse on a 
plane surface with two pins and a thread loop. Each ellipse has 
two focal points. 

We may take this spherical ellipse as the base, and the center 
of the sphere as the vertex of an elliptic cone, a pair of which 
will work in true rolling contact when the 
vertices are at a common point, the center of 
the sphere, and the axes taken as lines radiat- 
ing from the center of the sphere through the 
focal points above mentioned, one in each el- 
lipse, and at an intervening angle equal that 
expressing the length of the major axis of the 
elliptic cone base ; similarly as axes are taken at 

focal points A and B in Fig. 41. „ v <? , 

. Fig. 84. 

The wheels may be finished to the spherical 

form in the same way as illustrated in Fig. 83, the drawing being" 
made on a wooden spherical blank or normal sphere, as drawing- 
board, as in Fig. 84. 

On a great-circle arc of the sphere assume the portion FADG 
as the major axis of the proposed spherical ellipse, making FA = 
DG, so that A and D may serve as focal points. 

To find a point E in the spherical ellipse, take any arc FH in 
the dividers, and A with the focal point iasa center, describe an 
indefinite arc at E. Then with the remaining portion GH oi the 
major axis, and with center at the focal point D, describe an arc 
at E cutting the former one, thus determining a point E in the 
spherical elliptic arc. Find other points in the same way suffi- 
cient for drawing in the curve GEF, the half of the elliptic curve, 
required. 

In order that the elliptic wheels may have axes at right angles,, 
the angle FOG must be 90°; but the wheels and angles may be 
greater or less than this. 




66 PRINCIPLES OF MECHANISM. 

Taking the line A as an axis, this wheel will mate with an^ 
other like it. 

Instead of spherical wheels for this case, Professor MacCord has 
shown that plane wheels may be obtained by cutting the elliptic 
cone by a plane at right angles to the line 01, giving an outline on 
the plane which will not be a true ellipse, though approximating 
one. The contact point of the pair of these wheels will always be 
found on a fixed straight line connecting the axes, and perpendicu- 
lar to that which bisects the angle between the axes. 

The advantage of the plane wheels over the spherical ones is 
doubtful, since in laying out the teeth on these blanks for gear 
wheels the teeth are not normal to the edges of the wheels except 
at four points, while for the spherical ones they are normal 
throughout. 

As in the plane elliptical wheels of Fig. 41, a link may be 
mounted upon pins fixed at the focal points not occupied by the 
axes, as at D in Fig. 84 for wheel A. The pins at their bearings in 
the link must converge to the center of the sphere. 

2d. Multilobed Elliptic Bevel Wheels. — These may be treated by 
applying the process of Figs. 85, 86, and 87 to Fig. 44, etc. 

A pair of spherical elliptic unilobed wheels thus made have the 
same law of angular velocity as plane elliptic wheels of Fig. 44, as 
may be proved by aid of Figs. 85 and 86 ; while wheels of Fig. 84 
liave not. 

Third and Fourth. Parabolic and Hyperbolic Bevel 
Non-Circular Wheels. 

These can probablybe best treated under the general case where 
any pair of plane wheels can readily be turned into bevels. See 

Figs. 85 to 88. 

Fifth. Transformed Bevel Wheels. 

Having given a pair of bevel non-circular wheels, in the form 
as traced on the normal spherical segments, the wheels being sup- 
posed divided into elementary pairs of angles, arcs, and radii, they 
may be transformed in the three ways mentioned for plane wheels, 
viz.: 

1st. By multiplying any pair of elementary angles by the same 
quantity. 

2d. By adding a constant length of meridian arc to all the me- 




AXES MEETING. 

ridional radii of one wheel, the elementary perimetric arcs remain- 
ing unchanged. 

3d. By adding to one and taking an equal length away from 
the other of a pair of meridional radii, the perimetric arcs remain- 
ing unchanged in length. 

Similar Bevel Multilobes. — Thus the half-wheels of Figs. 83 or 
84 may be changed from 180° spherical sectoral wheels to 90° or 
60°, etc., by Rule 1st, just stated, and several of them combined in 
one wheel of several lobes, two like ones of which will work truly 
together. 

Dissimilar Interchangeable Bevel Multilobes.— By Rule 2d, Figs. 
83 and 84 may be changed from 180° sectors to 90°, 60°, etc., where 
the perimetric arcs, as well as the difference of the limiting spheri- 
cal radii, will remain unchanged in length. 

In this way a 2-lobed, 3-lobed, 4-lobed, etc., wheel may be found 
from Fig. 83, which will mate with 83or with each other. 

Likewise for Fig. 84. 

Partially Interchangeable Bevel Multilobes.— Starting with a 
pair of correct-working dissimilar bevel non-circular wheels as 
those of Fig. 78 turned into bevels, one of the pair may be trans- 
formed by Rule 2d into a series of spherical sectors of unchanged 
perimetric lengths, and difference of limiting radii; but with such 
angular widths as to add even to 360°; and likewise treating the 
other of the primitive pair, we have two series of wheels, any 
one of one series of which will work truly with any one of the 
other series, where no two of either series separately will work 
together. 

Bevel Non-Circular Wheels of Combined Sectors. — Here, as in 
plane wheels, sectors of wheels of different class may be com- 
hined into one wheel, and the several mating sectors into another, 
the new wheels mating correctly. 

Thus the half-wheel of Fig. 84 may be combined with the half- 
wheel of Fig. 83, the limiting radii being made to agree, and both 
constructed on equal normal spheres; the result being a pair of 
one-lobed wheels, one side being elliptic and the other the equi 
angular spiral. 

Again, each of the above 180° sectors may be reduced to 90° by 
Rule 1st for transformation of wheels, or one to 70° and the other 
to 110°, etc., and combined, making pairs of 2-lobed wheels. Simi- 
larly, pairs of 3-lobed, 4-lobed, etc., may be brought out, but they 
will not be interchangeable. 



68 PRINCIPLES OF MECHANISM. 

Interchangeable Multilobes and Unsymmetrical Lobes. — Half- 
wheels of Figs. 83 and 84 may be transformed by Rule 2d into 
sectors that may be combined, both kinds into a single lobe, pre- 
serving the perimetric arcs and limiting radii, several of which 
lobes constitute wheels such as to realize interchangeable multi- 
lobes. 

Velocity- Ratio in Bevel Non-Circular Wheels. — In Fig. 7 the 
velocity-ratio in circular bevel wheels equals the inverse ratio of 
the radii of the wheels, or ratio of the perpendiculars to the axes 
from the point of contact G. In this way in non-circular wheels 
we can find the velocity-ratio, or number of times faster one 
wheel revolves than the other at any instant. If the driver A 
revolves uniformly we can find the fastest and slowest motion 
for the driven wheel B; and that one is a certain number of times 
faster than the other is often all that is required, when the above 
wheels will answer the purpose. 



CHAPTER V. 
BEVEL NON-CIRCULAR WHEELS IN GENERAL. 

The special cases of bevel non-circular wheels above considered 
were those of the equiangular spiral and of the elliptic curve 
which could be readily studied in the bevel form, and for which 
oases the law of angular velocity was not important. 

But in practical problems in mechanism where the law of 
motion of the driven wheel is essential, either for a part of or for 
the entire revolution of the driver, the forms of the wheels cannot 
be assumed to be certain spirals nor ellipses, but be determined both 
as mates, to suit the stated law of motion, as was the case in the 
like problem for plane wheels. 

As the exact velocity-ratio depends upon perpendiculars upon 
the axes from the point of contact, and not upon meridian circle- 
arc radii, a complication enters by reason of which it is found ad- 
visable to work out the wheels as plane wheels first, and in con- 
formity with the stated laws of motion, and then to convert them 
into bevels. 

Therefore the wheels are first wrought out, as in Figs. 69, 75, or 
78, as plane wheels, after which bevel wheels, possessing the same 
laws of motion, are determined. 

Take Fig. 85 for the pair of plane wheels as worked out in 
strict accordance with the stated laws of motion, as in Fig. 69, A 
being the driver and B the follower. 

Divide the wheels into parts or sectors of 5° or 10° each, the 
dividing lines serving as radii for reference. 

In Fig. 86 draw the lines AO and BO with the same interven- 
ing angle A OB as that to be intercepted by the axes A and B of 
the finished bevel wheels. Draw A^B perpendicular to AO, 
where its length is equal the line of centers AB of Fig. 85. Also 
draw B X A perpendicular to the line BO, where its length is also 
equal the line of centers AB, Fig. 85. Then draw the line AB, 



70 



PRINCIPLES OF MECHANISM. 



Fig. 86. By this construction the triangle ABO is isosceles. Also 
draw the circle A'B' from as a center, to represent the normal 
sphere in section. 




Fig. 85. 



Then lay off perpendiculars AC, Aa, Ab, etc., Fig. 86, equal to 
the radii AC, Aa, Ab, etc., of wheel A, Fig. 85, and draw parallels 




FA'A, D A 
Fig. 86. 



AAA 



to A 0, as Cq, ar, bs, etc., giving points of intersection q, r, s, etc., 
on AB. Likewise draw perpendiculars BC, Bm, Bn, etc., Fig. 86 T 
equal to the radii BC, Bm, Bn, etc., of wheel B, Fig. 85, and draw 
parallels to B 0, as Cq, mt, nu, etc., giving points of intersection q, 
t, u, etc., on AB. 

Then draw the lines from the points q, r, s, t, u, etc., to 0, de- 
termining the intersection points C, a, b, m, n, etc., on the circle 
arc A'B' , representing the spherical blank in section. 

This completes the drawing preparatory to work on the spheri- 
cal blank in wood. 



BEVEL ^ON-CIRCULAR WHEELS IN GENERAL. 



71 




Let Fig. 87 represent the spherical blank for a pattern for the 
wheel A in plan and section, the radius AO being equal A' of 
Fig. 86 of the normal sphere. On the 
plan draw meridian lines with the same 
scheme of intervening angles CAa, aAb, 
bAc, etc., as in wheel A, Fig. 85. Then 
with compasses opened to the arc A'C, 
Fig. 86, place one point at A, Fig. 87, and 
strike an arc at C, cutting the meridian at 
the point C as shown. Then the point C, 
Fig. 87, is one point in the periphery of 
the bevel wheel A. 

Another point is found by taking A' a, 
Fig. 86, in the dividers, and laying it off 
on the meridian Aa of Fig. 87, determin- 
ing the point a of the periphery. Like- 
wise, using distances from Fig. 87, deter- 
mine points b, c, d, etc., on all the 
meridians of wheel blank A, Fig. 87, 
when the wheel outline may be drawn in and the blank cut down 
thereto. 

The sectional view of Fig. 87 shows how the blank can be cut 
away within the outline to a spherical web, or to the desired num- 
ber of arms, as for the pattern from which to obtain castings for 
wheel A. 

A second spherical blank for wheel B is also turned up with 
the radius AO, Fig. 87, equal A' 0, Fig. 86, as before, for wheel A. 

On this blank are drawn meridian lines with the same scheme of 
intervening angles as shown on wheel B, Fig. 85. Also on all the 
corresponding lines of the blank lay off the arcs B'm, B'n, etc., as 
in case of wheel A ; draw the perimetric outline, and cut the wood 
blank to shape as before, giving a wheel B to mate with wheel A,. 
as a pair of patterns for castings in pairs, as the final result of the 
problem. 

The wheel outlines thus found are of course usually the pitch 
lines for non-circular gears, the construction of teeth for which is 
explained later. 

In the above the wheels were treated as one-lobed. But either 
or both may be of several lobes, or one of a pair may be internal. 

Usually the angle A OB, Fig. 86, is a right angle, and AB — AB 
(of Fig. 85) V2. 



72 PRINCIPLES OF MECHANISM. 

The proof of the above figures and process was not attempted 
with the explanation. To show it to be the correct one to carry 
forward into the finished bevel wheels the same law of angular 
motion as the plane wheels of Fig. 85 possess, we note that the 
velocity-ratio for the wheels, as shown in Fig. 85, is 

Angular velocity of A _ BC 
Angular velocity of B ~ A C' 

In Fig. 86, if OA and OB be axes, we find that for an arm A C 
attached to axis A, operating an arm BC attached to axis B, by 
having the ends joined together at q, with the arms suitably located 
on the axes, the velocity-ratio will be 

Angular velocity of OA _ Eg _ BC _ GC 
Angular velocity of BO ~ Dq~ AC '~ FC 

= velocity-ratio of the finished wheels for contact C, since FC, Dq, 
etc., are all perpendicular to the axes of Figs. 86 and 87. 

The same is true of all other pairs of radii. Hence the bevel 
non-circular wheels thus obtained have the same laws of angular 
motion as the plane Wheels of Fig. 85, as required. 

Fig. 88 is an example in wood for patterns for castings, of a 





Fig. 88. Fig. 89. 

pair of bevel elliptic unilobed and bilobed wheels, laid out by the 
above process. 

In Fig. 89 we have a photo-process copy of a pair of bevel non- 
oircular wheels in metal, laid out by above process from Fig. 73, 
consequently possessing the same law of angular velocity. The 
wheels are about 7 inches between centers, and at 45° angle of 
intersection of axes. 



BEVEL NON-CIRCULAR WHEELS IN GENERAL. 



73 




Fig. 90. 



XAYING OUT BEVEL NON-CIRCULAR WHEELS DIRECTLY ON THE 
NORMAL SPHERE. 

1st. One Wheel Given, to Find its Mate.— Let A CbD represent 
the given or assumed wheel traced free hand upon the normal 
sphere, the pair of wheels to be 
complete and not sectoral. 

Assume B for a trial center for 
wheel B, and the arc ACB on a 
great circle of the sphere for a line 
of centers. Assume points a, b, c, 
etc., and find mating points m, n, o, 
etc., such that Cm = Ca, mn ■== ab, 
etc. ; also spherical arc Bm + arc 
Aa = arc ACB; arc Bn + arc Ab 
= arc ACB, etc., when the peri- 
metric curve for wheel B may be 
traced through the points C, m, n, 
o, etc. 

If B was assumed too far from A the wheel B will have a gap 
in it, when another trial point B may be assumed and the work 
repeated till the wheel B closes. 

2d. Laws of Motion Given, to Find the Wheels. — Let Fig. 91 
represent the actual normal sphere, -and assume the wheel cen- 
ters A and B on it or axial lines A '0 
and B' at the desired intervening angle, 
say 90°. 

Lay out around A the auxiliary sectors 
with angles to represent the motion of A> 
the sectoral angles being passed over in the 
movement of the finished wheel A in equal 
times. Likewise draw the mating auxiliary 
sectors around B as shown. 

To find a pair of mating points a and m 
in the peripheries of the finished wheels, 
make the arc A'li = Ae, and B'i = Bf; find 
h by parallels to A' and B'O, and draw 
Fig. 91. Qty. Then make Aa — A'j, and Bm = B'j, 

giving a and m for the desired mating points. 

Thus proceed with all the auxiliary sectors, when the wheels 
may be drawn in through the points a, b; m, n, etc. 




74 PRINCIPLES OF MECHANISM. 

This will make the velocity-ratio of the finished wheel the same 
as that of the auxiliary sectors rolling upon each other, because the 
velocity-ratio for the points a and m in contact at C is the same as 
for the sector Ae rolling on the sector Bf, or the same as for axis 
A'O to drive axis B'O by contact of ends of perpendiculars Ih and 
ri\ hence 

Velocity-ratio =s — = -=^ = '-f-, 
J n kt js 

so that the finished wheels have the desired velocity-ratio. 

These wheels will each turn through a pair of mating angles in 
equal times. 

If A revolves uniformly, as is usually the case, the sectors for A 
will be equal to each other. 

Similar Wheels, Lobed Wheels, etc., may be worked out on 
the normal sphere, similarly as for plan wheels in Figs. 79, 80,, 
and 81. 



CHAPTER VI. 

Case III. Axes Crossing without Meeting. Skew- Bevel 
Non-Circular Wheels. 

For this we will attempt only the general case, or the counter- 
part of non-circular bevels as brought out in Figs. 85, 86, and 87, 
as pertaining to non-circular skew bevels; wooden blanks being 
here presumed also, as receiving part of the work of the draftsman* 

Fig. 92 is a pair of circular hyperboloids answering for the case 





Fig. 



Fig. 93. 



of pitch surfaces for circular skew-bevels, given to aid in acquaint- 
ing the mind with the forms we have to study. 

In the present case the rolling or pitch surfaces are non-cir- 
cular hyperboloids with fixed axes, and a given shortest distance- 
between them. 

75 



76 



PRINCIPLES OF MECHANISM. 



At C the surfaces do not coincide, but lie at an angle with each 
other so that if extended beyond C they would intersect, as in 
Fig. 93 at F. This figure may be taken to represent a pair of 
blanks from which we could cut a pair of non-circular skew-bevels. 
These should be turned in a lathe, of nearly spherical form, such 
that their line of intersection AFB as seen in plan is a circular 
line with as its center. 

The line AFB, in space, will be so nearly a circle that it will 
be assumed to be one, and a common meridional line of the con- 
vex surfaces of A and B, so that DAH and EBG will be equal 
spherical surfaces intersecting in the line AFB> made to a radius 
greater than A to be determined. 

The final result in the laying out of these wheels will be the 
non-circular outlines on the blanks A and B, to which the material 
may be cut, to realize the final rolling wheels. 

As a first step in the process of obtaining these outlines, deter- 
mine a set of non-circular wheels with axes parallel, as in Fig. 85, 
that will work in conformity with the laws of motion as required 
ior the skew-bevels. 




Fig. 94. 

Also draw Fig. 86 from 85 as before, making the angle A OB 
<equal the angle AOB of Fig. 93, as the given angle between the 
axes in the plan of the skew-bevels; and the circle A'CB' ', Fig. 86, 
with the radius AO, Fig. 93. 

Then draw Fig. 94, where the portion AOB is the same as Fig. 
86, and where 0A X and OB 1 is the horizontal projection and KB aad 
the vertical projection of the axes, LO being the ground-line. 



NON-CIRCULAR SKEW BEVELS. 7? ; 

In the vertical projection the common perpendicular between. 
the axes is OR, projected in plan in the point 0. 

The line of contact between the rolling surfaces is here, in ele- 
vation, a shifting line PN, always parallel to KR and LO, and in. 
plan it is a shifting radial line OP, PN and OP being the projec- 
tions of that line as in one position. 

The circle AJ)B X is taken as the projection of the line of inter- 
section AFB of the extended wheel-bh i;£a shown in Fig. 93, A x O 
being the distance of the pole of wheel A from the common per- 
pendicular between the axes, B x being the like distance for wheel 
B; which distances are equal A f and B' 0. 

The line of intersection A 1 PB 1 of the extended wheel-blanks is 
seen in elevation in the line OPK. 

To correctly draw the line, divide the arc A X DB X into equal 
parts by points i,j, k, I, etc.; project lines up from these points to 
KR; then extend KR to F, where the angle FOB f = angle A x OB x ; 
draw the semicircle FeO, which intersect at e, f, g, etc., by radial 
lines from to points of equal division in the arc B'UVE, and 
project lines from the points e,f, g, etc., over to intersect the lines 
previously projected up from A 1 B 1 to KR. Through the proper 
points of intersection G, H, etc., draw the line KPHGO for the 
correct line of intersection sought. This may be regarded as a line 
on a vertical circular cylinder A X DB X with center at 0, which, de- 
veloped, would show the line OPK as slightly S-shaped. 

In the geometrical work this line will be used in its true form ; 
but for turning up the w r heel-blanks an approximating circle arc 
will be used, found by revolving OPK around axis to the hori- 
zontal plane when the point K describes the arc KL, falling at L, 
and B x falls at I. To draw the circle arc A X JI a center 8 is found 
on the line A x O extended, in such position that A X S = 18. 

The circle IA X T is a meridian arc to which the non-circular 
skew wheel-blanks are to be turned in the lathe in order that their 
line of intersections, as in Fig. 93, may be such as to run through 
AFB, as required. 

In Fig. 95 we have the projection of the wheel-blank A turned 
up to fit a circle pattern cut to the circle IJA X T, Fig. 94. On this 
blank lay off the angles CA a, a Ah, etc., as a copy of the angles sim- 
ilarly lettered on the drawing of the plane wheel, Fig. 85, previously 
explained, drawing the radiating lines or meridians to the edge of 
the blank, as shown in Fig. 95. 

To find the point C on the wheel-blank radius AC, take the 
length Cx, Fig. 94, in the dividers, and, without changing, place. 



78 



PRINCIPLES OF MECHANISM. 




one foot at P on the curve OPK at such position that the other 
foot falls at N on the line OR, making PN perpendicular to OR. 

Then, retaining the one foot of the di- 
viders at P, swing the dividers, and open 
them till the other foot falls at 0. Retain- 
ing this length in the dividers, place one foot 
at J on the circle A^JI while the other foot 
falls on Afi at Y, making JY — OP in the 
diagram and perpendicular to Afi. Then 
swing the dividers around J and open them 
till the other foot falls at the point A r With 
this length A , J in the dividers lay off A C, Fig. 
95, on the meridian A C of the wheel blank A. 
To find the point a, Fig. 95, take the dis- 
tance az, Fig. 94, in the dividers, and apply it 
on another line PN, PO, JY, JA X , as before, 
and finally Aa, Fig. 95, and so proceed for 
all the points in the periphery of A, when 
the outline of the wheel A may be drawn in. 

Otherwise explained, lay off Cx, Fig. 94, from R on RK. Pro- 
ject the end of this line down to P. Revolve P to Q about 0. 
Project Q to J. Then AJ = AC on wheel-blank Fig. 95. Pro- 
ceed likewise for other points. 

For wheel B, as for A in Fig. 95, take distances from C, m, n, 
etc., to line OB' , Fig. 94, and apply these as before from R toward 
K. Project to P, revolve about to Q; project Q to J, and take 
JA X to apply on wheel-blank of wheel B in the manner indicated 
in Fig. 95 for meridian distances PC, Bm, Bn, etc. 

When ready to cut the edge of wheels A and B to the perimetric 
lines just laid out, a question will arise as to what direction CD or 
bevel to give across the edge of the wheel, as it will not always be 
normal to the spherical outer and inner surfaces, and the right-line 
elements of the surface CD will be askew with the sphere normals. 
One way to proceed is to cut the minimum non-circular section 
of the wheel from a piece of plane thin material, or thicker with 
the edge bevelled, as EF, Fig. 95, and to mount this upon an axial 
rod, FA, extended from the wheel blank A, Fig. 95, making 
AF = OA,, Fig. 94, or UA, Fig. 93. 

To lay out the small wheel EF first draw a series of diametric 
lines with the intervening angles the same as in the upper part of 
Fig. 95, or in the plane wheels of Fig. 85. The lengths of the 



NON-CIKCULAK SKEW BETELS. 79 

radii will be the several heights ON, Fig. 94, which may be noted 
as the lengths Cx, az, etc., are placed in the several positions PN 
.as explained. These heights ON are to be laid off on the proper 
radii of EF, Fig. 95, in order; when the outline may be traced in, 
and the section EF cut to shape. The radial lines on EF, as well 
as on A, should be plainly marked and lettered, or numbered, to 
distinguish them. In mounting EF upon the rod the mean radius 
approximately should be selected and noted, as, for instance, that 
corresponding with AC in the figures, and this radius on EF 
placed at an angle with the corresponding radius on wheel, which 
-angle is PON, Fig. 94. 

Then in dressing off the edge of wheel A, Fig. 95, use a thin 
straight-edge, or thread across from EF to A C, trained upon cor- 
responding points of the two, cutting through from C to D till it 
coincides with the straightened thread stretched from E to C, and 
likewise for the several points of radii noted, when the intermediate 
portions can be cut by eye. 

Wheel B is to be treated in a similar manner. 

This gives us the rolling or pitch surfaces of the wheels. 

Instead of using the small wheel EF, Fig. 84, as a guide in 
stretching the thread to use while trimming off the material at CD 
and other points, it may be advisable to use a second blank like 
AC, turned to the same spherical surface except it may be thinner 
by cutting away on the concave side. Let us call this blank M. 

On. the spherical convex side of this second blank M we are to 
lay out meridians, straight or slightly curved, as explained for Fig. 
95, but with intervening angles in the inverse order of those of 
Fig. 84. On these meridians the same radial distances A C, A a, 
etc., are to be laid off also in the inverse order. 

This wheel, cut to line with quite a relief bevel, is to be mounted 
upon an axial rod like AF, Fig. 95, extended to twice the distance 
AF, or to twice OA lt Fig. 94; and at the intervening twist angle 
between corresponding radii of twice PON, Fig. 94; and with the 
concave sides of the blanks toward each other, and twisted in the 
right direction. See Fig. 139. Then in dressing off the blank at 
and along CD attach the line to the opposite blank M, at the cor- 
responding point C, and draw the line straight, and apply to CD as 
often as desired while dressing away the material, until coincidence 
is obtained entirely across CD, making this one right-lined element 
of the surface. If we call this element CC, because connecting the 



80 PKINCIPLES OF MECHANISM. 

points and C of the blanks, the next point may be aa, and the? 
next bb, etc. See Fig. 95. 

After dressing the blank across at CD, move the line and dress- 
ing action to the next point aa, finishing which as before; move 
again to bb, etc., for the whole periphery of A. On the portions 
of surface of the edge of A between the line elements CD, etc., as 
above found, the material may be cut away by eye, or if preferred 
the line may be stretched to intermediate points. 

Proof for the construction of the line OPK, Fig. 94, as de- 
scribed, is found in the formula expressing the proper division of 
the common perpendicular OR, Fig. 94, or A' B' , Fig. 10, into 
segments or gorge radii, in accordance with the velocity-ratio, viz., 
Fig. 10, calling angle A' CO' = a, B'C'O' = /3, A'C'B' = a, and 
the common perpendicular = c. 

O'C tan a = A'0' = x; O'C tan j3= B' 0' = y; 

tan tx x 

whence : -5 = - ; also x 4- y = c and a -f- 6 = a. 

tan fi y u 

-_,..,. , Q ± tan (a - «) c — x 

Eliminating y and p, we get f = , 

tan ol x 

which mav be reduced to x = c— cos (a — a), 

J sm a x ' 

in which, for Fig. 94, c = OR, a = A OB = A 1 OB 1 = BOB; 
a = any angle, as B'OV, and x = the corresponding height HZ. 

Thus OF = OR secant ROF = c . cosec a = ; 

sin a 

Of = OF cos VOE = -A- cos (a -a); 

HZ- OFsin B'OV- -A- cos (a - a) sin a = x. 

sin a 

Hence it appears that the height HZ, Fig. 94, is 'the result of 
the graphic construction of the above formula, and that the posi- 
tion of the point of contact or intersection of the line of contact 
with the line OP is correctly determined for each set of radii of 
the plane wheels, and that consequently the radii of the wheel 
blanks, made out as described, are correct. 

To be strictly correct, instead of meridians A C, Aa, Ab, etc., Fig. 
95, as mentioned, those lines as seen in projection should be laid off 
as copies of the line OPK as seen in projection in Fig. 94, and at 
intervening angles, the same as before. Then FE, Fig. 95, should 
be located with respect to a particular point P and angle PON. 



CHAPTER VII. 

NON-CIRCULAR WHEELS FOR INTERMITTENT MOTION. 

These wheels might properly have been considered under the 
cases of axes parallel, axes meeting, and axes not parallel and not 
meeting, but as their peculiarities in other respects are greater, it 
was thought advisable to treat them together, in deference to that 
refinement of classification. 

They are here classified in two ways, with reference to the de- 
vices for stopping and starting the driven wheel, viz.: 

1st. Where easement spurs are used together with locking arc 
of greater radius than that of the rolling arc; and 

2d. Where the locking arc is smaller than the rolling arc, with 
easements upon the ends of the pitch lines. 

1st. Where Easement Spurs are Employed. 

In Fig. 96 we have an example of the first kind, where ease- 
ment spurs or prongs are used for stopping and starting the 
driven wheel with reduced shocks. Here the arc /"m^\ 
DEF is non-circular and rolls correctly on the / \ „ a\ 
arc GMH. the arc from near D and F through G \^^r = f^\(i 
and H being circular and serving simply as a lock / ^%& \ 
for wheel B while standing still during its period /co~~^?;gC r6~d> ) 
of rest or intermitted movement. F \ aa [*>■ 

The curved piece IJ fast upon A, and KL fast V / 

upon B, have initial contact at the points /and K; ^T^^I 
when A is rotated so that /approaches K, and when 
/is carried down near to A, the quickness with which B is started! 
is reduced, as well as the shock due to it. Hence for rapid mo- 
tions AI should be short, while for slower speeds it may be greater.. 
In some cases IJ may be a mere pin, as in Fig. 11 for circular roll- 
ing curves. 

In either case the shape of DG and saddle curve GH should work 
with but little backlash when I J and KI are in running contact. 

The spurs here, as well as in Fig. 12, may work by rolling 

81. 



82 



PRINCIPLES OF MECHANISM. 



or sliding contact, though the latter will be found most convenient 
to lay out, and no more objectionable, unless it 
be in the matter of lubrication. 

2d. Where Locking Arcs and Easements are 
Employed. 

In Fig. 97t the wheels have the locking arc 
of smaller radius than the rolling arc DEF, in- 
stead of larger as in Fig. 96. The shapes of D 
and F will be much the same as in Fig. 15. 

The rolling arcs in Fig. 97, as well as in Fig. 
96, may have any non-circular form desired, ac- 
cording to requirement of practical problems. 
As in circular wheels, these may be made with axes meeting or 
axes crossing and not meeting, according to principles laid down in 
the proper places as pertaining to these problems. 

Fig. 98 is a photo-process copy of a pair of wheels like Fig. 96, 
turned into bevels, with axes meeting at 60°, 7 inches between 
centers, and of very considerable eccentricity, as best shown in the 
second part of Fig. 98. 




Fig. 97. 




Fig. 98. 

The process of Figs. 85 to 87 was followed in laying out these 

wheels. 

DIRECTIONAL RELATION CHANGING. VELOCITY-RATIO 

CONSTANT OR VARYING. 

ALTERNATE MOTIONS BY ROLLING CONTACT. 

Pitch Lines for Alternate- Motion Gearing. 
Among these movements there are those that are necessarily 
limited in extent of movement, and others not. 



^on-circular wheels for intermittent motion. 83 

First. Limited Alternate Motions. 
Velocity-ratio Constant. 
In Fig. 99 the driving half- wheel A turns continuously in one 
direction while the driven piece B reciprocates, and hence the 
change in directional relation. 




Fig. 99. 

In the present case the half-wheel is circular, and rolls on the 
straight line DE for movement in the one direction, the line DCE 
heing equal the semicircumference HCI. The arrangement is to 
be such that as the driven piece B, in moving to the right, reaches 
that position where H comes to contact with the extremity D, the 
other end / of the half-wheel arc is just on the point of taking 
contact at G, so that by further motion of A the rolling is trans- 
ferred from DE to FG and so continues with B moving to the left, 
till the last point, H, of the wheel makes contact at F, when the 
contact is retransferred to DE, and the motion again reversed. 

It is plain that the greatest extent of movement of B is the 
length DE — GF = HCI, and hence the movement is termed 
"limited." 

Also it is plain that if A revolves uniformly, the motion of B 
will be uniform and the velocity-ratio constant. 

Fig. 99 is simply the theoretical representation of the move- 
ment. In practice, gear teeth, as well as suitable positive engag- 
ing and disengaging devices, are to be applied, by which the move- 
ment will be still further limited, which gearing and devices will 
be treated under the proper headings. 

Velocity-ratio Varying. 

In Fig. 100 the driver A is a 180° sector of a log. spiral and, 
as before, rotates continually in the 
same direction, while the point of 
contact shifts over from the straight 
inclined line DCE to the parallel line 
FG, simultaneously with which the 
direction of motion of B is reversed. FlG - 10 °- 

If A revolves with uniform velocity, the movement of B is 




84 



PRINCIPLES OF MECHANISM* 



hastened as the contact at C approaches D or F, so that the ve- 
locity-ratio is a varying quantity. 

Drawing a line from the center of A through Cin a direction 
perpendicular to the slide bearings BB, we have the line of cen- 
ters, the center of motion of B being at an infinite distance from 
A, since the line DC is an arc of a log. spiral with the pole at 
infinity. 

The velocity-ratio is the same as given in connection with Fig. 
27, the rate of motion of B being the same as that of the point C 

as if revolving about A with the 
angular velocity of A. 

Another example of velocity-ratio 
varying is given in Fig. 101, where 
A may be taken as a portion of any 
one-lobed wheel, and the curve DE 
or FG found such as will mate with 
it. The same formula for velocity-ratio applies as in Fig. 100 
or 27. 




Fig. 101. 



Second. Unlimited Alternate Motions. 
Velocity-ratio Constant. 

The Mangle Rack. 

In Fig. 102 A acts by rolling contact against the bar BB 
secured to the flat surface of DE, in which latter a groove, FG, is 





O 
O 


1 






D 


F 


" K 




C 


5» 


-^7^ s ™ 


-rSp 


( 




H 

[G. 102. 





cut into which the end of the shaft of A is inserted, that A may 
be held in rolling contact with the barlike raised part BB. The 
ends of BB are rounded, equidistant from which the groove FG 
follows in its circuit about BB. The piece DE is fitted to slide in 
straight guides, so that as wheel A continues to revolve in the 
same direction, the bar BB and attached slide DE will be moved 



NON-CIRCULAR WHEELS FOR INTERMITTENT MOTION. 



85 



till A reaches the extremity of B, when further rotation of A will 
cause A to pass around upward at one end and downward at the 
other end, in continued revolving of A and reciprocation of BB 
and DE. 

A bar HK has a slot through which the shaft of A passes to 
prevent it from swinging to the right or left. 

This mangle-rack movement may be given a piece BB of any 
length, without limit. The part BB may be narrow, even reduced 
to a mere line. 

The velocity-ratio is the same as that in Fig. 27 or 100. 



The Mangle Wlieel. 

In Fig. 103 we have much the same 
construction as in Fig. 102, except 
here BB and FG are formed in a 
circle around 0, the part DE being 
here a circular disk mounted on a 
center at 0, so that the driven part 
BB and DE has an alternating cir- 
cular motion. 

The velocity-ratio is constant while 
B is moving in one direction, where 
BB is circular about also in the 
other, though in the second case it 
differs from that of the first when BB 
has a sensible radial thickness. 

Velocity-ratio Variable. 

The velocity-ratio will be variable 
when the piece BB in Fig. 102 is 
curved on a straight slide, or straight p IG 103 

on a curved slide, or with varied 
width. 

For Fig. 103 the piece BB may be of varied width, or non- 
circular with respect to the centre O, or in form of a spiral, even 
reaching to several turns about O, and with A non-circular, com- 
bined with the above modifications. 




86 PRINCIPLES OF MECHANISM. 



AXES MEETING. 

Bevel wheels in these movements are practical, and probably 
skew-bevels also, the construction of which must have due regard 

to the principles of bevels, or 



Tgf 



Fig. 104. 



skew-bevels, as laid down under 
rolling contact. 

In Fig. 104 is given a photo- 
process copy from a pair of 
mangle wheels under Axes Meet- 
ing at a right angle. The pitch 
line is merely a circle on a cylin- 
der set with teeth, which are sim- 
ply a row of pins with which the 
driver engages on either side. 



PART II. 

TRANSMISSION OF MOTION BY SLIDING CONTACT. 



CHAPTER VIII. 

SLIDING CONTACT IN GENERAL. 

In elementary combinations of mechanism acting by sliding 
contact, there are a driver and a follower with axes at a fixed dis- 
tance apart, the driver acting upon the follower at a point of 
contact at some distance to one side of the line of centers, in con- 
sequence of which there will be sliding of one piece against the 
other at the contact point, since rolling action occurs only when 
the contact is on the line of centers. 

In thick pieces the contact will be along a line of tangency, 
which line must be continually parallel to the axes when the latter 
are parallel, or meet with them at a common point when they are 
meeting. In each of the two pieces, all sections normal to the 
axis are similar figures. 

Velocity-ratio in Sliding Contact. 

In Fig. 105, take A as the center of rotation of a piece AJDK r 
with contact at D against the piece BLDM. Then when the piece 
AD is turned through some angle toward BD, the latter must turn 
also, when a sliding action will occur at the contact D. Hence 
AD may drive BD by sliding contact at D. 

To find the velocity-ratio for this turning of the pieces AD and 
BD about their centers A and B, while thus transmitting motion 
by sliding contact, find the center of curvature of the piece A JDK 



ss 



PRINCIPLES OF MECHANISM. 



or of the curve JDK at D, as some point E. Also find the center 
of curvature F, of the curve LDM at D, for the piece BLDM. 

Then DE will be the radius of curvature at D of the curve 
JDK, and DF the radius of curvature at Z> of the curve LDM ; 
and pieces might be cut from thin wood, one of shape AJDK, 
and the other of shape BLDM, which, when held by pins at A and 
B for axes, could turn by sliding upon each other in the manner 
proposed. For a small amount of movement, sufficient to deter- 
mine the velocity-ratio, the distance EF will be constant, it being 
the sum of the radii of the curves in contact at D. We observe 
that in this case E is a fixed point on the piece AD, and F a fixed 




Fig. 105. 



point on the piece BD. Now cut another pair of pieces, AFCG 
and BHCI, making E the center of curvature of the curve FCG 
at C, and F the center of curvature of the curve HOI at C, the 
point G being in this case chosen where the line EF intersects the 
line of centers AB. 

If now the two pieces A EC and A ED are fastened together 
with a common axis at A and coincident at E, likewise if the 
pieces BFC and BFD be united with a common axis B and coin- 
cident at F, and the two be mated on their centers A and B by 
axial pins and brought to running contact, then for a small move- 
ment the pieces will be in rolling contact at C, as in Fig. 2, be- 
cause this point is on the line of centers; and in sliding contact 
at D, as at first proposed; and the two actions will have a common 

7? C 
velocity-ratio which equals -— n . as in Fig. 2. 

A 

Therefore, to determine the velocity-ratio of a pair of pieces 

AJDK and BLDM transmitting motion by sliding contact, draw 

a common normal Z>C to the curves in contact, extending it to 



SLIDING CONTACT IN GENERAL. 89 

intersect the line of centers; then the velocity-ratio is given by the 
equality 

T_ BC 

v ~ a e 

where Fis the angular velocity of A, and v the angular velocity of 
1). Hence the velocity-ratio in sliding contact is the inverse 
ratio of the segments ^f the line of centers as the latter is cut by 
the common normal to the sliding curves. 

In Fig. 105 only a small movement was supposed to occur, 
beeause the curves in common tangency at D and C may not be 
circular to any considerable extent. But to realize an indefinite 
movement under like conditions, it is- only necessary to suppose the 
curves to be so shaped that the point of tangency G shall remain 
on the line of centers as for the non- circular wheels of Fig. 51, 
while the common normal EDF shall continue to pass through the 
same tangency point G For this, the centers of curvature E and 
F may be continually changing positions. 

The curves FCG and HC1 may be circles to the centers A and 
B, as in the pitch lines of circular gearing, when the point G of 
■the common normal CD becomes stationary on AB. 

SLIDING AND ROLLING CURVES WITH COMMON LAW OF MOVE- 
MENT NOT REQUIRED TO BE CONCENTRIC. 

In Fig. 106 let the curves AaCc and BbCd be a pair of correct- 
working non-circular rolling wheels in rolling contact at G ; and 




ADe and BDf a pair of pieces in action by sliding contact at D, 
and of such shape that the common normal at D, prolonged, shall 
-continually meet the point C. Then, because the velocity-ratio of 
the non-circular wheels is B Cover AC. and because the velocity- 



90 



PRINCIPLES OF MECHANISM. 



ratio of the sliding curves in contact at D is also BC over AC, it 
follows that DC must be a common normal at D, but not neces- 
sarily at C ; and that the law of the velocity-ratio is common to 
both actions so long as the curves are so shaped that DC is a com- 
mon normal at D and meets the point C. 

Fig. 107 is a photo-process copy of a model in metal of what is 
shown in Fig. 106. The lower sectors are the rolling arcs, and the 




Fig. 107. 



upper the sliding arcs. Their lengths are such that they all begin 
and end action together. The sliding curves cross the rolling curves- 
as in Fig. 108. 



SLIDING CURVES TO WORK CONFORMABLY WITH ROLLING CURVES. 

In Fig. 108 let AlCjck and BoCmgn represent a pair, or por- 




Fig. 108. 



tions of a pair, of correct-working rolling curves centered at A and 
i?. Assume a sliding curve eabcd, which may be a templet piece & 



SLIDING CONTACT IN GENERAL. 91 

cut to suit, and made fast to wheel A, the latter being also a tem- 
plet piece as shown to mate with a templet piece B. 

Find the mating curve iafgh, which, by sliding action upon 
eabcd, will transmit the same motion, A to B, as by the mutual 
rolling of the rolling curves or wheels A and B. 

Making Cm = Cj, mg =jc, etc., we will have mating points / 
and m, c and g, etc., in the peripheries of the non-circular wheels. 
That is, these points will come to contact in pairs, and in succes- 
sion, on the line of centers as the non-circular wheels roll in 
mutual contact. 

Following Fig. 106, we draw a normal Ca to the templet G, 
when a becomes one point in the required mating curve for B. 

From j draw the normal jb to the templet G. This normal 
makes a certain angle with the radius Aj. Then draw mf, making 
the same angle with Bm produced, and make mf = jb. Then 
when the wheels revolve till j and m meet on the line of centers, 
the line mf will coincide with jb, and the point / with b. There- 
fore,/ is a second point in the sliding curve for wheel B. Evidently 
g mates with c as a third point. Also h, i, and c are other points 
found in the same manner as was /; oi being equal to le, and the 
angle Boi being 180° — Ale. Hence the sliding curve iafgli may 
be drawn in, and a templet cut to the same, to be mounted upon 
the wheel or templet B. These sliding curves or templets thus 
made fast to their respective wheel templets A and B will evidently 
work harmoniously and agreeably throughout with the rolling 
curves or wheel templets A and B, and hence with a common law~ 
of angular velocity. 

The extent of movement of these sliding curves is often limited, 
but a series of pairs may be employed which engage in action in 
succession, as in the case of the teeth of gear wheels, where the 
rolling curves tangent at C answer for pitch lines, and the sliding 
curves tangent at a for tooth curves. We therefore next take up 
the somewhat extensive subject of 

THE TEETH OF GEAR WHEELS. 

The sliding curves or tooth curves brought out above are some- 
times called odontoids, from the Greek word for tooth. Also the 
rolling curves A and B in contact at C, the pitch lines in gearing^ 
are sometimes called centroids, from the fact that Wj, etc., are 
points of instantaneous axes of rolling motion of the describing 
templet of Fig. 109 to be described. 



92 PRINCIPLES OF MECHANISM. 

THE TRACING OF SLIDING CURVES, OR ODONTOIDS, BY MEANS 
OF TEMPLETS ROLLED UPON NON-CIRCULAR WHEEL PITCH 
LINES OR CENTROIDS. 

Templets are very useful aids in the study of or actual drawing 
of tooth curves for gearing, and we may employ pitch templets, 
tooth templets, and describing templets. Thus, a piece cut from 
thin wood to the form AlCjck, Fig. 108, may be called a pitch 
templet, and sometimes two pieces are prepared, one convex and 
the other concave, fitting each other along the pitch line ICk, 
either of which may be used at pleasure. Likewise, pitch templets 
may be prepared, fitting the pitch line OCmgn. Again, a piece, 
G, cut to the form of the tooth curve eacl, and another for the 
mating tooth curve, may be called tooth templets. Besides these, 
a describing templet may be employed which, in Fig. 108, rolled 
along on the curve or templet ICjc, will, by an attached marking 
point, trace the tooth curve eabc, or again, rolled on oCmg, describe 
the mating curve iafg. 

Having given the non-circular wheel pitch curve ICjc, to find 

the describing templet which, rolled along on the upper side of the 

pitch curve, will describe or generate the tooth 

curve eabc: Draw aj, aC, al, Fig. 109, of the 

same lengths as bj, aC, el, Fig. 108, and at 

intervening angles such that IC, Cj, etc., Fig. 109, 

shall equal the like lettered quantities of Fig. 108. 

Then draw in the curve ajCl, Fig. 109, and we 

have the figure of the describing templet sought, 

to complete which describing templet cut a piece 

of thin wood to the form of curve found and set a marking point 

at a. 

To trace the tooth curve eabc, Fig. 108, with this templet: 
Place the tracer, a, of the templet at c, Fig. 108, and proceed to 
turn the describing templet left-handed, causing it to roll along the 
pitch line jCl without slipping, when the points/, C, I, etc., of the 
describing templet, Fig. 109, will fall at the points/, C, I, etc., Fig. 
108, by reason of the equality of spaces, step by step; and the tracer 
point a will at like points of time fall at b, a, e, etc.. by reason of the 
equality of lengths of the lines ja, Ca, la, etc., Fig. 109, with the 
normals jb, Ca, le, etc., Fig. 108, and the equality of the angles in 
the two figures between arcs and lines at the points jCl, etc. This 
rolling of the describing templet may suppose that a concave pitch 




SLIDING CONTACT IN GENERAL. 



93 



templet fitting the pitch line cjCl is employed for the describing 
templet to roll upon, and the curve cae traced upon a plane which 
is fixed with reference to the pitch templet. In the illustrations,, 
comparatively few points, c,j, C, I, etc., are indicated, in order to 
secure clearness of the figures. 

For the same reasons that the tooth curve cbae is traced by 
rolling the describing templet, Fig. 109, on the pitch line cjCl, 
likewise the mating tooth curve gfai is traced by rolling the same 
describing templet on the pitch line gmCo. 

It is to be observed that the above describing templet applies 
for only the portions of the tooth curves of Fig. 108 which lie 
above or to the left of the pitch lines. For the portions cd and gh, 
at the right or below, a second describing templet, found as was 
Fig. 109, would generally be required where, as in Fig. 108, one 
full tooth curve, e to d, was assumed. In practice, however, instead 
of assuming the tooth curve, the describing templet may advisably 
be assumed, and of any preferred form; and the same is often 
applied on both sides of each pitch line. 



NAMES, TERMS, AND RULES FOR GEAR TEETH. 

As the names of the various parts of toothed gearing will be of 
frequent use, they are here given, in connection with Fig. 110, 




Fig. 110. 
for future reference. It will be a help also in the study of teeth 
that the proportions be familiarized. 



94 PRINCIPLES OF MECHANISM. 

A common rule which may be deviated from according to 
the judgment of the designer is: 

For Circular or Circumferential Pitch. 

Pitch = distance from center to center of teeth on pitch line, 

Addendum = 3/10 pitch; working depth = twice addendum. 
Dedendum = 4/10 pitch, or subaddendum. 
Thickness = 5/11 pitch. 
Space = 6/11 pitch. 

For Diametral Pitch. — For Cut Gears. 

Pitch = portion of diameter to each tooth, = diameter divided 

by number of teeth in the wheel. 
Addendum = one diametral pitch. 

Dedendum = one diametral pitch plus 1/7 diametral pitch. 
Thickness = nearly 1.57 diametral pitch. 



CHAPTER IX. 
TOOTHED GEARING IN GENERAL. 



DIRECTIONAL RELATION CONSTANT 

VARYING. 



VELOGITY-RA TIO 



TOOTH CURVES FOR NON-CIRCULAR GEARING. 

Case I. Axes Parallel. 

All correct working gearing of whatever name, circular or non- 
circular in outline of wheel, must possess tooth outlines which are 
describable by the principles indicated 
in Figures 108 and 109. 

The most general case is taken up 
here, instead of several proceeding 
from the simpler to the more general, 
because the general case is as readily 
comprehended as any special limited 
one. Thus, the whole general theory 
of tooth outlines for gearing will be 
mastered at once. 

Templets. 

The theory and practice for this case 
can probably be best explained by the aid 
of templets, a complete set of which, 
representing the general case of non- 
circular gears, is shown in Fig. 111. 
Portions of the non-circular rolling 
curves or pitch lines are shown as A ^ 

and B ; A 1 being a templet fitting the 
curve A on the inside, A 2 a second templet fitting it 




on the 

outside, both of which will fit together on the pitch line A. When 

95 



96 



PRINCIPLES OF MECHANISM. 



thus together, mark the point C, b x and b tJ thus made coinci- 
dent ; also B 1 and i? 3 are like fitting templets for the pitch line 
B, with points C, c x , c 2 marked. These marked points are all 
mutually mating points as between templets or pitch lines. These 
are the pitch-line templets, and, for brevity, may be called pitch 
templets. 

The smaller figures show the describing templets, aF and 
bG, with the marking point at a and b, and have their represen- 
tatives, in theory, in Fig. 109. All these templets are supposed 
to be cut from thin pieces of wood, metal, or other suitable 
material. 



USE OF TEMPLETS IK DESCRIBING TOOTH CURVES. 

In Fig. 112, the describing templet aF is shown as being rolled 
along the pitch templet i? 2 , while the marking point, a, is describing 

the tooth curve c x a ; also the same 
describing templet is shown as being 
rolled along the pitch templet A } and 
describing the tooth curve b x a. Now, 
if no slipping occurs in the rolling, and 
it be continued on both pitch lines till 
the arc c x E be made equal to b x D, then 
will D and E be mating points in the 
non-circular rolling arcs or pitch lines, 
and aE will equal aD both as to arc 
and chord, and the segmental figures 
aE and aD are equal in every respect. 
In this rolling, the tooth curves b x a 
Fig. 112. anc [ ca are supposed to be traced by 

the marker a, and if extended, would give the dotted lines above 
a, a. These may be supposed to be traced upon the same paper 
that the pitch lines are drawn upon, and that the paper is trans- 
parent. 

Now if we place one drawing upon the other so that they will 
be in common tangency with the points E and D coinciding, then 
the lines aD and aE will also coincide with the two points a in 
common ; and the tooth curves b t a and c x a will touch each other 
at a. The resulting diagram for this will be as shown in Fig. 113; 
the two figures of the entire describing curves, indeed, coalescing 
into one at aDEF. 




TOOTHED GEARING IN GENERAL. 



97 




The two tooth curves b x ad and c x ae will be tangent to each 
other at a, instead of intersecting; because it is evident that if the 
describing templet aDEF be rolled either e 

way on the pitch lines, the point a will move 
to lower positions when the rolling is on Ab x B 
than when on Bc x . Also, it is plain that the 
line aDE is a common normal to the tooth a, 
curves b x ad and c x ae, since the point DE' 
serves as a momentary center about which to Fig. 113. 

describe small portions of the curves b x d and c x e at #, as in describ- 
ing a circle about its center. These facts are confirmed by refer- 
ence to Fig. 108. 

We observe further that the tooth curves b x ad and c x ae are 
wholly above their respective pitch lines and the point of contact, a, 
to the right of the line of centers through DE. 

The principles brought out above being perfectly general, they 
embrace all possible forms of correct working teeth, such as epicy- 
cloidal, involute, conjugate, paraboloidal, octoidal, etc. 

Hence all correct working tooth curves must possess the follow- 
ing properties : 

1st. They must continually remain in mutual tangency. 
2d. The mutual tangent point of a given pair of tooth curves 
must remain on the same side of the line of centers and of the 
pitch lines. 

3d. The straight line joining the point of mutual tangency of 
the tooth curves and of the pitch lines 
must be a common normal to the tooth 
curves; and these normals should never 
intersect between the tooth curves and 
their pitch lines, and they must always 
intersect or be tangent to their pitch 
lines. 

As Fig. 113 provides for contact 
only on the one side of the line of centers 
and above the pitch lines, contact below 
and to the left may be obtained by 
using the same, or any other, describ- 
ing templet below and to the left, 
as in Fig. 114, where the other tem- 
plet, bG, Fig. Ill, is used, and rolled on 
A and B below as shown in Fig. 114 and combined as in Fig, 




Fig. 114. 



PRINCIPLES OF MECHANISM. 



115, making the starting points, Z> 2 and c a , coincide with 5^, 
Fig. 113. 

This work, being all similar to that of Figures 112 and 113, 
will not require detailed description. 

The Tooth Profile. 

When combined with the points bfi^, c x c^, in common at C, 
at the line of centers, and omitting the describing templets, we 
have Fig. 116, in which LCM is a complete tooth profile for 
one side of a tooth of the wheel A, and NCO a tooth profile 
for one side of a tooth of the wheel B. Now, as the wheels 
turn one way from the position shown, the tooth contact moves 





Fig. 115. 

out in one direction from (7, or the opposite direction for the 
opposite turning ; from which it appears that in continuous re- 
peated revolutions of the wheels A and B there will be contacts of 
the tooth curves as they approach the line of centers and also in 
recession therefrom. 

These tooth curves in full lines in Fig. 116, correlated with 
other suitable ones in dotted lines, will furnish the outlines of a 
pair of teeth as indicated in Fig. 116. 



POSITION OF TRACING POINT ON DESCRIBING CURVE. 

The curves b t ad and c t ae, Fig. 113, may be called trochoids 
when the curves A and B are not circular, the former being an 
epitrochoid and the other an hypotrochoid. 

When the tracing point, a, is not on the arc of FD, but 
within or without, the curve generated is called respectively the 
prolate or curtate epitrochoid or hypotrochoid. 

The tooth curves of Figures 113, 115, and 116 do not represent 
the most general case possible, since the points a and b are there 
carried on the curve, while it may be within or without. There- 



TOOTHED GEARING IN GENERAL. 



99 




Fig. 117. 



fore, let us examine the curtate and prolate curves in quest of 
advantages over those of Figures 113 to 116. 

1st. The Curtate Trochoidal Tooth Curve. 

Fig. 117 represents the case of the curtate curves, the tracer, 
a, being carried on an overreaching piece Ka, fastened to the 
describing curve FK, aK being normal to KDEF. 

Take L and N as mating points in the pitch curves A and B, 
at which K is placed in starting the tooth curves, or trochoids, 

Gla and HJa. Then LG will be on a 
normal to A, and UN on a normal to B, 
and it is found that the portion HJ of 
HJa will interfere with more or less of 
the curve Gla, showing that the portion 
HJ cannot be used as a tooth curve. 
With HJ removed, Gl may be also, as 
useless. 

These curtate trochoidal curves from / and J upward will 
serve for tooth curves, and also others below A and B, but they will 
be found seriously inadmissible from the fact that there will be a 
wide space on either side of DE within which actual working 
tooth contacts are impossible, the very space where such contacts 
are by far the most desirable and valuable. The limit of this 
space to the right of DE is found by placing the tracer point, a, 
at J, and then noting the distance DJ. This will be found to 
increase with aK, and vanish only as aK disappears. 

2d. The Prolate Trochoidal Tooth Curve. 

In Fig. 118, the tracer, a, is placed on the normal Ka, then with 
L and N mating points at which K starts, as KDEF rolls on A 
and B respectively, we will obtain the 
proposed tooth curves, Ga and Ha. 

These curves will not interfere and 
may, theoretically, serve for teeth, but, 
practically, the curves are so nearly par- 
allel to the pitch lines A and B that they 
will be more likely to slip and wedge the 
wheel apart with serious crowding, than 
to give desirable driving action. In other 
words, practical tooth curves should be 
perpendicular to the pitch lines rather 
than parallel to it, and hence the prolate trochoidal curves of Fig. 




100 



PRINCIPLES OF MECHANISM. 



118 are inadmissible, though less so the shorter aK is made until 
it vanishes. 

Hence the most favorable position of the tracing point or marker, 
0, is that it be exactly on the describing curve itself, as in Figures 
113 and 115. 

Form and Size of Describing Curve, or Templet. 

Modification of the tooth curve, or odontoid, to a certain 
extent may be effected by employing one or another form of 
describing curve. 

Thus, the describing curve may be a circle, ellipse, parabola, 
cissoid, spiral of many turns, or any arbitrarily assumed free-hand- 
drawn curve, each giving its own peculiar tooth curve when used 
on the pitch lines A and B ; the flank curves apparently varying 
more than the face curves, as due to the changes in the describing 
curve as illustrated by the use of a small describing curve, la', or 
large one, Ja", giving rise to the flank tooth curves Da' and Da" 
respectively. 

For gears engaging externally the faces will always be found 
as lying between the involute and pitch line, while the flanks may 
be anywhere from the pitch line on the one side to the pitch line 
on the other. 

Thus, in Fig. 119, aD is the involute, described by rolling the 
straight line Fa on the pitch line A, the mating pitch line, B> 

being in this case straight, as for a 
rack, in order that Fa may be used 
to develop the flank. 

The curves -above mentioned 
rolled along DF, starting the tooth 
curves at D, will evidently trace 
them all more or less below the in- 
volute DG y according to size, the 
smaller ones when re-entrant mak- 
ing one loop after another, accord- 
ing to number of convolutions on A. 
These curves will all be normal 
to A at D except those developed by spirals which have an infinite 
number of coils around the marking point a, as in the case of the 
logarithmic spiral S with a at the pole. Thus the so-called invo- 
lute teeth, with involute bases within A, have tooth curves which 
intersect A at an angle. Involute gear teeth, which at first seem 




Fig. 119. 



TOOTHED GEARING IN GENERAL. 



101 



to be an exception to the general theory of Figs. 113 and 115, are 
seen, nevertheless, to fall within the general theory, the tooth 
curves being generated with a log. spiral describing curve, rolled 
outside of A for faces, and inside for flanks, these flank curves 
being terminated exactly at the base circle, because here the radius 
of curvature of the log. spiral just equals that of the pitch circle A, 
upon which it rolls internally. 

Example of Individually Constructed Teeth. 

In non-circular gears, where the curvature of the pitch lines 
continually varies, no two tooth curves will be alike, so that every 
one must be described individually to be strictly correct. Fig. 
120 is an example of a carefully made drawing of such gears, for 




Fig. 120. 

which the describing templets used for every face and flank were, 
for convenience, all circles, but varied in size according to the 
above principles, to favor the shapes of the teeth to prevent their 
being too weak across their flanks at points of sharpest curvature 
of pitch lines. 

Non-circular Involute Gears. 

In this case, as well as in circular involute gears, it is most con- 
venient to make use of involute bases, or curves, a little within the 
pitch lines to which the tooth curves are involutes, as above sug- 
gested, instead of using rolling spirals. 




102 PRINCIPLES OF MECHANISM. 

Let Fig. 121 represent a pair of non-circular pitch lines to be 
set with involute teeth, and let the 
curves P, I, J inside of A, also R, L, M 
inside of B, be the bases of the invo- 
lutes. The pitch lines are in contact 
at C, and for this position of wheels 
the bases of the involutes must be such 
that a common tangent PR will pass 
through the point C, and not only so for 
this one position, but for all positions 
of the wheels throughout the full ex- 
Fm. 121. tent of their rolling, whether segmental 

or otherwise. 

The inclination of the line PCR is arbitrary, except that undue 
irregularities of the base curves may be avoided. In circular gear- 
ing the base curves are usually concentric circles, but here they 
will usually be nearest to the pitch lines at points of sharpest 
curvature. 

Probably the best way is to place the pitch lines in various 
positions of mating taugency, draw a line PCR for each, and then 
trace in the enveloping curves P, I, J for each wheel. Then the 
involutes SCDE and TCFG may be drawn for the teeth. It is 
seen that the involutes cannot extend within the base curves as 
at 8 and T. If the addenda call for more room, the space between 
teeth may be arbitrarily deepened below 8 or T to a suitable root- 
line depth; but proper tooth contacts cannot go below these points,, 
nor beyond the tangent points P and R, by reason of interference. 
These teeth are less advantageous for gears of considerable 
eccentricity than those of epitrochoidal form of Fig. 120, because 
of the greater crowding action explained in Fig. 128, but they have 
the advantage of requiring less attention for maintaining exactly 
the distance A B between centers. 

Conjugate Teeth. 

This practical method of constructing tooth curves where one 
of a pair is arbitrarily assumed, and the other determined from it, 
would seem to contradict the refinements of the above. theory to the 
extent that one tooth may be compelled to take any preferred form 
under sheer caprice. 

Thus, prepare the pitch templets A and B, Fig. 122, and fasten 



TOOTHED GEARING IN GENERAL. 



103 



to B a sheet of paper D, and to A a tooth templet EF elevated 
sufficiently above A to allow the paper D to go in between 
A and F. 

Then bring the two templets A and B into proper contact at C 9 
Fig. 123, and while held securely draw the pencil along the tooth- 




^___ 


-?V— . 






fV 

0~0| 


A 




Fig. 122. 



Fig. 123. 




Fig. 124. 



curve edge of the tooth templet FCF, copying its shape upon the 
paper D. Then roll A upon B some distance without slipping, as 
indicated by the dotted line at A', and 
while held fast, draw another line along 
the edge of FCF. Thus draw several 
lines for A in positions to the right and 
left, when D will present a series of lines, 
shown dotted in Fig. 124. Now, the en- 
veloping curve GCH, or full line just 
touching all the dotted lines on D, will be the best mating tooth 
curve for FCF. All the tooth curves for one complete wheel 
might be thus assumed, and the mating ones for the other wheel 
found in the same way. 

This process is seen, after all, to present nothing but what is 
disclosed in Fig. 108, since G there corresponds with F in Fig. 123, 
so that a similar process carried out in Fig. 108 would result in the 
curve hgfai. 

Limited Inclination of Tooth Curves. 

In the practice of designing non-circular gears of considerable 
eccentricity, it will be found very desirable at times to make the 
teeth nearly radial and oblique to the pitch curves, as in the full 
lines of Fig. 125, instead of normal to the pitch curves and in- 
clined to the radius; thus to prevent the tendency of the teeth to 



104 



PRINCIPLES OF MECHANISM. 



slip out of engagement, as well as the excessive crowding apart of 
the gears. 

Therefore, let Fig. 125 represent portions of a pair of rather 
eccentric non-circular pitch curves A and B, with a proposed pair 
of teeth E and G in contact at the line of centers C, which, in the 




Fig. 125. 

light of assumed tooth curves of Fig. 122, and in spite of the theory 
previously detailed, are set nearly radial to avoid crowding. 

Now, with B stationary, roll A around, without slipping on B 
to the dotted position A', with E following around to E' ', shifting 
the contact of the pitch curves from C to D. Then it will be found 
that the tooth E at E' fatally interferes with the tooth G, showing 
at once that the amount of departure from the pitch curve normal 
toward the wheel radii shown in the figure is absolutely prohibited. 
In some actual wheel designs where this partiality for the radius 
existed, the teeth necessarily were doctored before the wheels 
would work. 

Thus, in Fig. 126, we have a photo-process copy of a pair of 
gears made for practical use in machines, examples of which Avere 
exhibited at the Centennial of '76, in which an 
effort at inclining the tooth toward the radius, 
giving it a hooklike form, was evidently attempted, 
but was vetoed by the theory when the wheels were 
put to work, as is plain by the fact that the hook 
feature was removed from the hook or overhanging 
side. One interesting fact incidentally connected 
with these wheels is that they are cut gears, the 
gear cutter to cut them having twenty-five cutters. 
Thus we have a rare example of other cut non- 
circular gears than elliptic. 

Other examples of the endeavor to obtain hook- 
like teeth were found at the Centennial; one show- 




TOOTHED GEARING IN GENERAL. 



105 



ing this peculiarity in a moderate degree being illustrated in Fig. 
127. 

Another example was found in a 
shingle-sawing machine from Wisconsin. 

In all the above discussions, from Fig. 
92 on, we have treated the case as if the 
wheels were plane figures without thick- 
ness, while in reality we may suppose them 
to have any arbitrary thickness as cylin- 
ders, transverse sections of which form the above figures. 




Fig. 127. 



Nearest Approach of Tooth Curve toward the Radius. 

To determine absolutely the nearest approach of the tooth curve 
to the radius, we have only to refer to the principles laid down as 
following Fig. 113, and especially that of the intersection of the 
pitch line by all the normals to the proposed tooth curve. Thus 
the curve cj), Fig. 114, will become more and more erect, with? 
respect to the pitch line BE' , the farther the intersection E' is 
kept away from the point c 2 : and this is favored by a large curve 
bG, the largest being the straight line. But this same curve must 
roll inside of A, and hence the curve A itself is the largest possible 
one. For this, however, the flank b 2 b would be merely a point, and 
hence objectionable from considerations of wear. 

It therefore appears that a sharper curve than A is desirable. 
In Fig. 128, A and B is a pair of non-circular wheels where the 

attempt is to be made to carry 
the tooth curve GC as far as prac- 
ticable over toward the direction of 
the radius BC, changing GC to some 
position NC as dotted. 

First, drawing a normal NO to 
the proposed dotted curve CN, this 
normal is found to escape the pitch 
curve OL altogether, and hence the 
curve CN\s steeper than admissible, 
since the normal must intersect the 
pitch curve CL. 

Next, we propose a describing 
curve, E, of large radius, such as 
will but just go within the mating 
pitch curve CM and roll, giving the shortest admissible flank CI. 




106 PRINCIPLES OF MECHANISM. 

This describing curve E must be accepted as that which will give 
the greatest permissible deviation of GC toward NC. 

This tooth curve GC is, however, far from coinciding with the 
direction of the radius BC, and hence as the flank CI drives against 
the face GC, there will be a more or less serious crowding apart of 
the wheels A and B. 

Wear in practical use of wheels is the criterion by which CI is 
limited in length, it being, at best, much shorter than CG against 
which it works. Good judgment would protest against CI being 
reduced to a point, as above suggested, for the case where E con- 
forms with CM\ and also the figure shows that this only very 
slightly moves the curve CG from its present position toward CJ\ T . 

Hence, though the utmost possible careening of CG toward CJV 
occurs when C'/is reduced to a point, yet practical considerations 
favor a slightly less elevated curve CG, to prevent an appreciable 
length of flank CI ; and we conclude that a tooth curve CG, lying 
near the radius BC prolonged, is impossible. 

On the other hand, the tooth curve may be inclined the opposite 
way, or toward the pitch line, to any desired extent, as shown in Fig. 
128 at DH&nd BE; it being simply necessary to make the describing 
curve, for example FK = JP, quite sharp in curvature, or small, 
and a spiral of several turns about the polo J for inclination of 
tooth-curve at D. 

Practical Limit in Eccentricity for Non-circular Gears. 

In theory there is no limit short of a radius for each wheel; but 
in practice friction interposes a limit, the full investigation of which 
is out of place here, so that a few examples, only, will be cited to 
indicate how essential are the principles of Fig. 128 in gear 
designing. 

Fig. 129 shows a pair of gears made for use on a machine to run 
at about 350 revolutions per minute. They are 4-j inches between 
centers and have involute teeth, both being cast in iron from one 
pattern and to work without lubrication. They were laid out after 
the principles of Fig. 79. 

When started in the newly developed machine, whole segments 
of the rims of these gears would drop out upon the floor in less 
than two minutes, though believed, as far as ordinary theoretic 
strains were concerned, to be ample in strength. On examination, 
it was found that the involute tooth curves were too much inclined 
to the pitch lines, so as to actually block the wheels at positions 



TOOTHED GEARING IN GENERAL. 107 

where the radius was also much inclined to a perpendicular to the 
rim. 




Fig. 129. 

It was found that elliptic wheels would give a more favorable 

obliquity of pitch line to radii, and that teeth generated by large 

describing curves like E, Fig. 128, would give easier action; where- 

^ ^ upon new wheels were adopted 

A^^Jm £^^n^9 with these changes and with coarser 

^^^^B B^P W&P teeth, a photo-process copy of a 

-•.. KgS i^jf fefe P a ^ r o;t ' wn i°h i s given in Fig. 130. 

^k ^^^ In these wheels every effort was 

fl K^^H^^^^V made to provide against breakage 

w w W by "blocking," two considerations 

Fig. 130. being included besides those above 

mentioned, consisting, first, in thicker teeth on the sides of the 

wheels than at the ends, and, second, shorter teeth, compared with 

pitch, than for usual proportions. The teeth are also much coarser 

than in the previous wheels of Fig. 128, so much so that but 

little room was left for arms and openings. 

A quite notable case of very eccentric gears is given in Fig. 
131, made in cast iron, 2 feet between centers, the larger being 
some 3£ feet long, and used on a hay-press, where the larger wheel 
is driver. Here the teeth are of unequal size as well as in Fig. 
130, the proximity of the axis in the small wheel controlling the 
size at the end. The tendency to block in action is apparently 
great, as due to the unusual length of the teeth, they being here 
extended to sharp points. 

An examination of the figure indicates that for the large wheel 



108 PRINCIPLES OF MECHANISM. 

as driver the blocking tendency at places is very great, so much so 
that thorough lubrication must be a necessity to bring the coefficient 
of friction down to permit the wheels to work at all. 




Fig. 131. 

Fortunately for this case the wheels are used in a horse-power 
machine, where, if the wheels block from deficient lubrication, the 
horses may stop as a sign for lubricant wanted, the wheel having 
sufficient strength to resist the horses. In a machine with the 
inertia of a heavy fly-wheel or other parts involved, the result would 
be different. 

The full line drawn as a common normal to a pair of teeth just 
taking contact, with the large wheel driving right-handed, strikes 
very near the center of the driven wheel, making a case of positive 
block with coefficient of friction at 0.083. As the coefficient of 
friction is about 0.12 to 0.16 for dry cast iron, and 0.05 to 0.08 for 
lubricated, it is clear that these gears cannot run except with teeth 
well lubricated. 

In the case of the dotted line for the next tooth there is still 
greater doubt. 

If these teeth were cut off to the ordinary addendum of 3/10 
pitch, the tendency to block would be very much less,, and still bet- 
ter if cut to the minimum where one pair of teeth engage in action 
as the preceding pair quits engagement, and the wheels would then 
probably run without lubrication. 

Whence it appears that unduly long teeth, such, for example, as 
run to a point, are especially objectionable in very eccentric non- 
circular gears. 



TOOTHED GEARING IN GENERAL. 109 

Again, where one of a pair is always to be driver, the addendum 
of the driven wheel may with advantage be made unduly short with 
that of the driving wheel increased. 

In heavy gearing rigidly connected with heavy revolving parts, 
the ratio AQ over QH, Fig. 128, where QH is a normal to CH at 
H, should not exceed 0.4 to 0.5 for teeth running without lubrica- 
tion. 

It therefore appears that there is a practical limit of eccentricity 
in non-circular gearing, occasioned by the " blocking " tendency at 
points where the pitch line departs farthest from a normal to the 
radius, and that this limit is favored by short addendum of teeth 
of driver; by large tooth-generating curve for side of tooth in ques- 
tion; by thick and stout teeth; by lubrication; and by giving the 
driver a slightly greater pitch than follower. 

Substitution of Pitch-line Rolling for Teeth in Extremely 
Eccentric Gears. 

In some cases of very eccentric gears, where blocking would 
surely occur with teeth, the latter may be omitted with advantage 
and the driving action from driver to fol- 
lower effected by the mutual rolling of the 
pitch lines. 

Fig. 132 is an example of this, which 
represents the limiting case of the "quick 
return" movement, where a crank and pit- 
man is made to drive the connected slide 
forward at a uniform velocity and back in 
much less time, or "quick." The model 
of Fig. 132 represents the theoretical limit 
of uniform motion forward from beginning 
to end, and back in an absolute instant, or 
no time at all. This is evidently impos- p lG 133 

sible, since the driven wheel must make a half-turn in the absolute 
instant, during which the driver would be stationary with no power 
to drive. 

In the model this difficulty is removed by placing the handle 
on the driven- wheel crank or lower wheel in the model. 

In the absence of teeth, the eccentric pitch surface of the 
driver acting upon that of the follower can drive in one direction, 
but not in the other, and to prevent derangement some device 
must be supplied. 




110 



PRINCIPLES OF MECHANISM. 



In elliptic gears a link may be added as in Fig. 133, wher.e 
the eccentricity is too great for good working teeth throughout, 
so that a pair, only, of teeth is added at the ends and the link 
introduced to hold the elliptic pitch lines in 
rolling contact. 

But in Fig. 132 a link cannot be employed, 
so that a circular sector with non-circular arcs 
at the ends is added to engage with a mating 
piece on the opposite wheel. These are so 
shaped as to hold the upper wheel stationary 
for the allotted time of the half-turn of the 
lower wheel, after which the non-circular 
curves keep the rolling pitch-line arcs in con 
tact for the space where teeth are omitted. 

Fig. 134 is a carefully made drawing of a 
pair of gears used for a while in a certain 
screw machine for a " quick return" to dirve 
a slicL uniformly forward and back with the least allowable time 
greater than an absolute instant. It is produced, in effect, by 
modifying Fig. 132, as by splitting the long projecting point of the 
nppcr wheel and spreading it to DAE by a circular sector, mated 
with FCG in the other wheel, the two wheels being otherwise modi- 
fied in a manner to preserve the quick-return principle exactly. 
As adopted, the advance occupies 3/4, and the return 1/4 of the 
time. The line of centers is about five inches. 

From H to F, and G to i, the pitch lines roll upon the mating 




Fig. 133. 




Fig. 134. 



pitch lines DJ and EK, respectively; teeth engaging for the re- 
maining arcs. 

To prevent the gears from getting deranged in this case, pro- 
jections from A, shown by dotted lines at N and 0, are added, 



TOOTHED GEARING IN GENERAL. 



Ill 



which fall inside of projections L and M f respectively, and serve 
to keep the pitch lines approximately in contact, and assuring the 
engaging and disengaging of teeth for several hundred revolutions 
per minute in either direction. 




Fig. 135. 



Fig. 136. 



Fig. 135 gives a better view of the projections L, M, N, and 
of Fig. 134. 

These wheels served with entire satisfaction, as operating gears, 
for a considerable time in the machine they were designed for. 
But as the return, though decidedly more moderate than in Fig. 
132, was yet decidedly too quick for the purpose, other wheels with 
still less rapid return were found advisable, and obtained in the 
wheels shown in Fig. 136. 

Here the return was so extended that one of the rolling arcs 
was dropped, the other being necessarily re- 
tained to preserve the law of uniform motion 
of the slide, the one set of projections L and N 
being also retained to provide against mis- 
engagement of teeth. 

These wheels, as in Fig. 136, are doing good 
service in extended use and are thoroughly 
practical gears. 

Internal Non-circular Gears. 

These are practical for complete wheels 
where the " internal " gear or that having teeth 
within the rim is at least twice as large as the 
pinion, or smaller wheel; and the tracing of the 
teeth is as before except for the fact that 
generally both pitch lines are convex in the same direction as in 
Fig. 137, so that a further description is unnecessary. 




Fig. 137. 



CHAPTER X. 
TEETH OF BEVEL AND SKEW BEVEL NON-CIRCULAR WHEELS. 

Case II. The General Case of Axes Intersecting. 

We here assume that the pitch lines are already made out act- 
ually on the normal spheres, or else as ordinates, angles, etc., by 
which the curves may be drawn, as explained under Case II of 
Rolling Contact, with Axes Meeting. 

Let Fig. 138. represent a pair of such rolling wheels with cen- 
ters at A and B, and with C the common tangent point. Suck 




Fig. 138. 

wheels with axes meeting are conical as explained under rolling 
contact, and may extend in thickness to the vertex of the cones. 
The front surfaces of the wheel at AC and CB are spherical sur- 
faces, being the center of the sphere and the common vertex of 
the cones. The finished wheels are usually as if cut from spherical 
shells as in Fig. 83. 

The describing curve or templet to carry the tracing point is 
here a cone, also, with base aCF, and with vertex at in common 
with the other cones. The tracer or marker is here a line aO. 

112 



TEETH OF BEVEL AND SKEW BEVEL NON-CIRCULAR WHEELS. 113 

Let L and N represent mating points on the rolling cone-wheel 
bases, LC and JVC being equal. Then placing the line aO of the 
describing cone coincident with the element LO of the wheel 
cone, and rolling it around to the position aCF, we describe a sur- 
face LaOL, which may be extended to c as shown. Also placing 
the same describing cone inside the wheel B, and with the line aO 
coincident with the element NO of the wheel B and rolling it. 
around to the position aCF, we generate the surface NaON, which, 
may also be extended by further rolling. These two generated 
surfaces will be tangent to each other along the element aO, and 
hence these surfaces form proper tooth surfaces for teeth of these 
wheels, whatever their thickness. 

In the case of spherical shells, the tooth curves may be traced 
directly on them, both sides if desired, by means of rolling templets 
of wood cut concave to the spherical form and of any. preferred 
outline aCF. 

In Fig. 138 we have a face for a tooth of A, and a flank for a 
tooth of B, contacts between which will always be inside of B and 
to the right of A CB. Contacts on the other side may be obtained 
by rolling the templet a OF, or any other one, inside of A and out- 
side of B, thus procuring a flank for .4 and a face for B. 

For non-circular wheels with large teeth it will be necessary for 
accuracy to trace every individual tooth curve for the entire wheel,, 
no two curves being alike; but the same templet will not be re- 
quired except for the pair of tooth curves that are to work to- 
gether, as La and Na in Fig. 138. 

In small teeth less care is needed and approximate tooth lines 
will serve, as, for instance, for 1 or H inches pitch or less. In 
some cases, as for instance in Fig. 98, where the gears were cast, 
the patterns had the rim formed to the root line, when teeth 
blocked out for the full depth in separate pieces, by a drawing 
giving some excess of curvature at the sides, were fastened upon the. 
rim and the patterns finished and the gears cast. 

The Tredgold's approximate method, hereafter described fully 
in connection with circular bevel gears, may be employed here with 
sufficient accuracy for any practical purpose. 

Involute teeth may be laid out by first finding the base lines and 
using rolling planes on the base lines, similarly as lines are used in 
Fig. 121, the tooth surfaces being generated by a line on the rolling 
plane running to of Fig. 1 38. 



114 



PRINCIPLES OF MECHANISM. 



Case III. The General Case of Axes Crossing Without 
Intersecting. 

non-circular skew bevels. 
A method for the exact construction of teeth for this case is 
not known, nor even for circular skew bevels for serviceable gears, 
except such as are taken from near the shortest common perpen- 
dicular between the axes. 

Suppose the pitch curves have been determined upon the spher- 
ical blank as at p, Fig. 139, by Figures 93 to 95. Then the adden- 
dum and dedendum lines ma} 7 be laid out as at d and e, Fig. 139. 
Between the last-named lines the tooth curves are to be drawn as 
shown by the method of Fig. 138, rather small describing curves 
being used to prevent the teeth from interfering unduly by reason 
of the admitted approximate form of these teeth for non-circular 
skew bevel. 

Then, as explained in connection with Fig. 95, construct the 

second spherical blank M to mount 
upon an axial rod equidistant with it 
from the gorge, the teeth being laid 
out upon M in the inverse order with 
respect to A; or in like order as the 
blanks are viewed in Fig. 139, the 
convex side of A and the concave 
side of if being towards the eve. 

Cut the teeth on M to line and 
bevel back towards the concave side 
so that the line or string CC, drawn 
tight, will spring from the line of the 
tooth curve on the convex surface el 
M. 

Then the blank i may be cut to 
the addendum line and to the tooth 
curves by stretching the string 
across, at various corresponding 
points of A and M, as at A, m, n, o, etc., and at as many inter- 
mediate pairs of points as desired. 

These teeth will be twisted, but the face and flank surfaces will 
be composed of straight-lined elements. 

Instead of cutting these teeth from the solid material of the 
blank A, the latter may be dressed off carefully to the dedendum 




Fig. 139. 



TEETH OE BEVEL AND SKEW BEVEL NON-CIRCULAR WHEELS. 115 

line and separate blocks for teeth may be prepared and fastened 
upon the blank, and finished subsequently unless quite small and 
short along CD. 

The wheel i?, mating with A, maybe made in a similar manner. 
If they are of the same size and alike, one pattern for casting will 
serve for both A and B, but otherwise not. The wheels may, either 
of them, be made 1-lobed, 2-lobed, etc. 

When a pair of wheels are thus far completed they will not run 
together smoothly because of the already-mentioned incorrectness 
of the tooth outlines for this case. 

The running conditions may be improved by mounting the 
wheels as mated in mesh, and by revolving slowly under careful 
scrutiny, the high or interfering points noted, and dressed oft' by 
hand, this process being continued till the desired degree of smooth- 
ness of running is obtained. 

The resulting wheels, obtained as above, may be of wood and 
used for patterns for casting in metal with no greater further 
expense than for circular wheels. 

For wheels of one inch pitch or less it is probable that teeth 
may be formed in separate pieces and made fast upon the blanks A 
.and B, and dressed to the dedendum line and surface. 



CHAPTEE XL 

INTERMITTENT MOTIONS, 

TEETH; TOGETHER WITH ENGAGING AND DISENGAGING 
SPURS AND SEGMENTS IN GENERAL. 

TEETH. 

We now come to consider the teeth and spurs of intermittent 
motions whether having circular or non-circular pitch lines. 

The teeth present no new problems over segmental or partial 
gears, as, for instance, in Fig. 96, the pitch line FED when 
equipped with teeth is as a segmental gear, to mesh or mate with 
the segmental gear GME, and any preferred form of tooth may be 
adopted and laid out as already explained. There should, however, 
be a space to start with at F and D and a tooth at G and H, as here 
shown in Fig. 140 at N and K. 

Directional Relation Constant. 

THE ENGAGING AND DISENGAGING SPURS. 

The spurs GF and HI are to start the driven member B from 
the full stop on the locking arc into full motion, as for the initial 
engagement of the teeth, and this usually by contact action of the 
spurs. Where a sufficiently large arc of movement can be allowed 
to A in which B is to be started or stopped, the spurs may work by 
rolling contact instead of sliding. 

In Fig. 140 we have a correct drawing of a pair of non-circular 
plane wheels of the intermittent order, showing two complete sets 
of prongs or spurs for engaging and disengaging. One set, num- 
bered 1, works by sliding contact both ways, and the second set, 
numbered 2, works by rolling contact; the same arc of movement 
ab for wheel A being required to swing B from rest into full en- 
gagement when the sliding-contact spurs are employed as when 
the rolling contact spurs 2 are in use. The same is true for the 
spurs on either side of the wheel, the arc cd being greater than ab, 
as required by the greater drop MN in radial distance. 

116 



NON-CIRCULAR INTERMITTENT MOTIONS. 



117 



The spur HI mates with FG, and they are so made that when 
the latter approaches the former, F and H are the first points to 
touch; this same contact starting B towards engagement of the 
gear teeth, when further movement causes the point of contact to 
move along toward G and i, arriving at which the full engagement 
of the gear teeth should be assured. 




Fig. 140. 

Where this movement takes place the circular locking arc JM 
must be cut away as at JK, and to just such extent that the wheel 
B shall not have undue backlash, it being prevented by the bearing 
of B upon JK for the one direction, and by the contact between 
FG and HI for the other direction. The same, of course, is true 
of the other side of the wheel A, where B' is shown as bearing 
against MN for the one direction of motion of B' and against the 
spurs 11 for the other -direction. It is essential for the satisfactory 
performance of these wheels that these curves and spurs be well 
mated with respect to reduced backlash. 

For rapid rotation the spurs need to be much the longest. For 
slow motion a pin may do for FG, with HI cut shorter to match, as 
shown at D and F, Fig. 141, where the shock is some 4 to 6 times as 
great as in Fig. 140. But the shock is materially reduced by giving 



118 PRINCIPLES OF MECHANISM. 

the point of the spur a curve as shown in the example at E and H 9 , 
Fig. 141, the intensity of the shock being judged of by the perpen- 
dicular distance from A to the normals shown. Some shock will 
occur at all events when the points H and F, Fig. 140, meet in 
contact unless HI and FG are so much extended that the normal 
FD to GF meets the center A. To determine the amount of 
shock in a given case, we are to consider the segments AC and bC 
of the line of centers Ab at the initial contact of H upon F, these 
segments being formed by the common normal FD. 

In EP and QR, Fig. 140, we have a pair of spurs working by 
rolling contact instead of sliding. These differ from the sliding 
spurs in being much longer for a like tendency to shock, and con- 
sequently objectionable, though the rolling action is in their favor 
in respect to wear. To determine the lengths for equal shock we 
note that the distance AE for the rolling spurs should be the same 
as the length AC for the sliding spurs. The same is true for the 
terminals of the other spurs at SX and VT. 

The rolling spur QR, which mates with EPK, is unavoidably 
concave because the contacts which occur on QR must be progress- 
ive from Q toward R, and must always be on the line of centers for 
rolling action; and for a gradual start of B from rest, the line QR 
near Q must be nearly radial and recline more and more with re- 
spect to a radius as shown by the line QR, and, in fact, should run 
into the pitch line of B as a continuous rolling line. Likewise 
EPK should run into the pitch line KN. 

Again, that the acceleration be normal, the curve EP should 
gradually turn over into the pitch line for the teeth at K, as if it 
was the termination of that line toward A, both portions EP and 
KN being, in fact, a continuous rolling arc, and, reasonably enough, 
should begin at E near A, and by continuous smooth curvature* 
pass the points P, K, and on to N. Thus this half of these wheels 
resembles the half of the wheels of Figs. 132 and 134, where a pair 
of rolling curves run into the pitch lines of gear teeth. 

At MN the locking arc is necessarily flexed to a greater dis- 
tance toward A, by reason of the non-circular pitch line NX em- 
ployed, and MN\& shaped as due to an effort to give B a quick 
move from lock to engagement, as may sometimes be necessary in 
the application made of these wheels. To this end, it is here found 
necessary to give B a more rapid motion for a part of this turn 
from rest to engagement, than it will have while engaged with the 
teeth. This is not so evident from the sliding spurs S and Xas 



NON-CIRCULAR INTERMITTENT MOTIONS. 119 

i'rom the rolling ones TTOVa'nd VW, for which the law of angular 
velocity is made the same. Examining the latter, we note that for 
contact of VW on TU from Tup to £7 the acceleration is rapid, 
and that at U the speed of B is full double that due to contact at 
N when the gear teeth have fairly engaged. One effect of this is 
to leave the tooth Y behind with liability of its interfering with the 
tooth it is approaching. Again, it appears prejudicial to accelerate 
the motion of B to 100 per cent, in excess and retard it again, the 
only excuse for which in practice is probably to be found in a 
necessity for a quick movement of B from rest to the full engage- 
ment of the gear teeth, or for a comparatively small arc cd for it. 

At all events it is certain that the rolling spurs TUN and VW 
are utterly impracticable at least from U to N, where the driving 
action of the spurs would be negative. 

That rolling spurs may be made practical in place of TUN 
and its mate, Fig. 140, we infer from the curve EP, that instead 
of going outward at first to U and then turning in toward A to N 9 
it should start at once from T on a curvature turning toward N. 
But an examination of the case will readily show that this is im- 
possible unless the arced is increased. 

In Fig. 141 the problem of practicable rolling spurs for wheels 
A and B is undertaken, where NBV is the total angle through, 
which B must turn in going from rest to engagement, and NAd r 
the corresponding angle for A. 

The angle NBV is divided into a series of decreasing angles 
from N toward V, then the same number of angles are laid off 
from NA toward Ad, all of which span slightly more arc on the 
radius ^i^than does the first of the series NV on the radius BN. 
Then the curves are drawn in according to the method of Fig. 69,. 
giving the rolling spurs NUT and NG V. 

It is plain that the portions of the rolling curves NG and NU 
are so nearly perpendicular to the radius as to be likely to work bad 
in practice unless provided with teeth, becoming part of the toothed 
segment, and that the easement curve is very much prolonged as 
compared with that of Fig. 140. Also it seems that the speed of 
B in some cases must be accelerated beyond that of the gear teeth 
in engagement and then retarded to the latter velocity. 

The arc of engagement e'd' is here very wide, as due first to thie- 
very considerable drop in the radius from AH to AN, and, second,, 
to the fact that AN is so short compared with AB. 

It appears, then, that rolling arcs may always be adopted for 



120 



PRINCIPLES OF MECHANISM. 



'b\ 



engaging or disengaging spurs of intermittent wheels in place of 

sliding arcs for the same; but 
that the former are subject to 
some prejudicial features not 
inseparable from the latter, but 
which outweigh the objection 
to sliding contact action of the 
latter. 

These wheels may be made 
as bevels for the case of axes 
meeting, a complete example 
in metal being shown in Fig. 
98, which worked satisfactorily. 
In skew-bevels they are 
doubtless possible, but proba- 
bly after a tedious time in 
construction they would be 
found unsatisfactory mainly 
by reason of the sliding along 
the line of contact of the 
various surfaces, requiring the 
spurs to be very thick, etc. 

In Fig. 140 the wheel A is 
shown with only one locking 
arc, and B one locking seg- 
ment. But either may have 
two or more, involving, how- 
ever, no new problems over 
these already considered, there 
being a toothed arc followed by 
a locking arc, and the latter 
again by a toothed arc, etc., in 
succession, spurs being provid- 
ed throughout. 

In case of slow movements 
of this kind the piece FG or 
S in Fig. 140 may be made 
shorter, indeed reduced to a 
mere pin as at E and 77, or F 
Fig. 141. and 7>, Fig. 141, thus simplify- 

ing the movement in construction. 





NON-CIRCULAR INTERMITTENT MOTIONS. 121 

By giving the spurs a very considerable curvature, as at E or a 

H, the shock due to contact of spur and pin will be materially 

reduced as compared with that attendant upon the spurs and pins 

at i^ 7 and D. Here the normal EC or HC is to be drawn, and the 

segments ^4 6 Y aud bC measured by which to judge of the shock at 

contact of pin and spur. The shock will depend upon the velocity- 

A C AC 
ratio at the instant of initial contact, that ratio being -j—j or -7777. 

Comparing with Fig. 140, it seems possible that pins and spurs may 
be so made that the shocks are no more prejudicial than in Fig. 140. 



SOLID ENGAGING AND DISENGAGING SEGMENTS OF 
INTERMITTENT MOTIONS. 

Intermittent motions with spurs attached are objectionable in 
coarse and heavy machinery from the tendency to loosen and be- 
come detached. These wheels, of solid casting complete, are much 
the more satisfactory in such places as binders for reaping ma- 
chines, etc., both for simplicity and cost, especially where they are 
small or run slowly. By suitable shapes of the engaging and dis- 
engaging segments, however, they may run at speeds equalling 
those provided with spurs, as in Figures 140 and 141. (See Fig- 
ures 15 and 16.) 

A suggestion as to the shapes of the locking segment of B is 
obtained from Fig. 141 at E and H, where the spurs might seem- 
ingly be in the plane of the wheel B, and A correspondingly cut 
away to a circle arc coinciding with the prongs of B as proposed, 
the latter so connected as to form the locking segment for B. 
This would lead us at once to Fig. 15. Following the suggestion, 
we obtain Fig. 142 as a pair of these wheels of the non-circular 
form, B being at rest on the locking arc. 

In designing these wheels, the distance IL and LE should be 
sufficient to prevent the tendency of the ends of the prongs / and 
iTfrom biting into the arc DLE and blocking the movement, or, in 
case of heavy wheels, causing breakage. To examine into this, we 
will find that the angle BKM must exceed the so-called " angle of 
repose " for the material and conditions at K. A portion of the arc at 
J should be cut away to prevent " clinging " to the arc IK upon 
the terminal points D and E of the arc DLE. 

Between EG and DF a portion is cut away to allow the prongs 



122 



PRINCIPLES OF MECHANISM. 



of B to pass as the latter is started into motion. These gaps are 
undercut at G and F to reduce shock due to initial contact with 
the prongs of B, the intensity of which shock may, as before, be 
examined relative to the distance AP or AQ from the center of 
motion of A to the normals at F and 67. This shock is zero when 
AP and. AQ are zero. 

No direct and simple rule of graphics is known for laying out 
the wheels of Fig. 142, beyond that for the toothed pitch lines, 





Fig. 142. 



Fig. 143. 



where the arc FNG must equal RT8. The radius AL is arbitrary, 
except that it ought to be less than either terminal portion of the 
toothed arc FNHG. After assuming a trial arc DLE, and with 
trial curves for the wheel A at i^and G sketched in, a circle may 
be drawn through B as at rack for the latter and a trial templet in 
cardboard cut out by guess for BIK, with a hole having a point at 
B f or center. This templet with guess forms at JR and KS is to 
be tried at several positions as shown at EGH, the center always 
remaining on the circle arc through B. Likewise at FD. A guess 
shape of the prong of B at / may be tried, and if found unsuitable 
a modification suggested by this trial may next be cut out and 
tried. By this tentative process the most suitable shapes at JR 
and KS and at DF and EGH are to be determined. The curves 
FHG and RTS must be correct rolling arcs for gear teeth. Sev- 
eral trials will probably be needed to bring out the most satisfac- 
tory shapes at F, I, TTand G. 



ALTERNATE MOTIONS. 123 

For slow movements, where the shock of starting and stopping 
of B due to speed is at a minimum, the prongs / and iTmay be 
less hooking and simpler, as in Fig. 143, for which, at a given 
speed, the shock is about four times as great as for Fig. 142. 

The intensity of shock may be examined into here by drawing 
the normals FP and GQ and considering the lengths of perpendic- 
ulars from A upon these normals. These are seen to be greater 
in Fig. 143 than in Fig. 142 and hence the greater shocks. 

Bevel wheels after either variety, Fig. 142 or Fig. 143, may be 
readily worked out according to Figs. 85 and 87 after the plain, 
wheels are drawn, as illustrated by Fig. 98 of a 
working model in brass. Skew bevels are here 
much more practicable than in the wheels of 
Figs. 140 and 141, and may be worked out ac- 
cording to Figs. 93 to 95, and be practical in 
running machinery. 




COUNTING WHEELS. 

These wheels, of high durability, may be con- 
structed as in Fig. 144, which is a positive 
movement with no possible likelihood of de- 
rangement except by breakage. For counting, FlG -^ 
the upper wheel should have 10 teeth. The 
" Geneva stop " is a familiar example of this class of wheels. 

ALTERNATE MOTIONS. 
Directional Relation Changing. 

The pitch lines for these wheels have been treated in Figs. 99 
to 103. 

The most essential features for successful wheels of this kind 
consist in the peculiar shapes of terminal teeth, or of segments, or 
of spurs, or of shifting devices, as in mangle wheels. 

These wheels may be divided into two distinct classes, viz., as 
those admitting of limited movements only, and those of indefinite 
extent of movement, to be taken up in succession. 

First. Limited Alternate Motions. 

The movement is in this case limited because the driven piece- 
must unavoidably make a forward and return movement for each 
revolution of the driver. The pitch lines of these wheels have- 



124 



PRINCIPLES OF MECHANISM. 



already been considered in Figs. 99 and 101, and the shapes of 
teeth later. 

But the most important consideration in devising these move- 
ments is that they shall have no positions for skips, derangements, 
etc., and that the continued action is absolutely positive through- 
cut. Also that in high speeds or in heavy movements of this kind 
the driven piece should be given a retarding motion in approaching 
the stop, and an accelerating movement when it moves off from the 
stop point, to avoid shocks and breakages. 

Here, as in the intermittent motions of Figs. 140 to 143, there 
may be employed engaging and disengaging spurs and also seg- 
ments. 

In Fig. 145 we have an example where spurs and pins are used 
in which A is the driver revolving either way continuously, and B 




Fig. 145. 

the follower sliding in guides D and E forward and back in repeti- 
tion, a complete movement both ways being made in each revolu- 
tion of the driver A. 

The wheel A and rack frame B are of the same thickness, with 
teeth to engage each other for the principal movement, the correct 
construction of which has already been explained. 

In the wheel A, at F and G, are pins to be engaged between the 
spurs FJ and GK made fast to the side or sides of the frame and 
used to insure the transfer of engagement of teeth from the one 
side to the other side of the rack frame BB. 

The spurs are here made to conform with the cycloidal curves 
GL and FL in order that the motion of B, when the pins move 
outward in their slots, may be such as to allow the teeth to go into 
engagement without interference. The spurs-? 7 and G may be ex- 



ALTERNATE MOTIONS. 



125 



tended as far as desired short of interfering with shaft A, but the 
spurs J and JTmust be cut away so as to allow the pins to enter 
the slots, just as they necessarily would if there were nothing but 
pins and spurs. In this example J and K are cut quite abruptly, 
K most so; by reason of which quite a shock will occur when the 
pin G enters its slot and strikes the spur, the intensity of which 
may be judged of by taking account of the length of the perpen- 
dicular AH from the axis A to the normal HG to the spur. The 
normal FI at F makes the perpendicular distance AI relatively 
very short, from which it appears that the shock due to contact of 
the pin at F would be very slight. 

To reduce the shock at G, the spurs may be extended to near 
M, where the normal to the cycloidal curve GML passes near to A. 
This, however, seriously shortens the segment FG of the segmental 
wheel A, greatly reducing its number of teeth. The movement 
would thus be much better adapted to high speeds, and judgment 
is to be used in designing to obtain the best adaptation to circum- 
stances. 

Assuming the velocity of A constant, the velocity of movement, 




Fig. 146. 

of B will be variable, and always proportional to the distance on a 
perpendicular to BB from A to the intersection with the normal 
to the spur at G while the latter bears on the spur and to the 
pitch line while the teeth are in engagement. 

This eccentric form of A for alternate motions is objectionable- 
on the score of ease of movement, and may be avoided except where 
a varied motion is essential. In Fig. 146 the variation of motion 
is less objectionable than here for a durable high-speed movement. 



126 PRINCIPLES OF MECHANISM. 

In Fig. 146 we have an example where solid engaging and dis- 
engaging segments instead of pins and spurs are adopted with 
which to reverse the movement with certainty and positively. As 
in Figs. 142 and 143, no pieces are attached by screws to become 
loosened, occasioning derangement. 

At reversal of motion the first driving contact occurs at F or 
G, according to direction of motion. Normals at the contact point 
being FH or GI, the shock due to contact will be proportional to 
the length AD or AE, as the case may be, DAE being perpendicu- 
lar to the direction of motion of BB and corresponding to the line 
of centers. 

The contact points F and G are put as near to the center A as 
possible to minimize the shock due to initial contact, immedi- 
ately following which the contact moves outward to the pitch line 
of the teeth when the latter engage in action and continue the 
movement. 

This movement gives the driven piece B the most rapid motion 
when near the middle of this stroke, the rate of motion being for 
each instant always proportional to the length of the perpendicular 
from A to the contact on the pitch line for that instant. 

The ends of the first teeth of both A and B are cut to circle 
arcs about the center A and of some breadth of bearing, to prevent 
the disabling of the movement unduly soon by wear of those parts. 

The hub of A is made to fit closely into the extreme portions 
of the opening of B to facilitate bringing J. and B into the correct 
relative positions when the contacts between teeth are shifted from 
side to side. 

Any non-circular form may be given to A, as preferred; though 
it will be found, on trial of several various curves, that it is favor- 
able for smooth running to have both the curves draw in toward 
A as in Fig. 146, instead of the contrary, as in the case of one side 
of A in Fig. 145. 

These wheels may be made with conical form of A and of B, 
the cones having a common vertex more or less remote from A, and 
the forward and back motion of B occurring as if swinging about 
an axis through the vertex of the cone and perpendicular to the 
axis of A. 

Also Figs. 145 and 146 may be laid out with BB on a circu- 
lar curve and the same be put into the spherical form, though a 
considerable cut and try work, with templets, will probably be re- 
quired before the final adopted figures are brought out. 



ALTERNATE MOTIONS. 127 

The Office of Motion Templets. 

In all this work of determining wheels and curves, as in Figs. 
140 to 146, templets will be found most useful. They may be 
formed of paper or cardboard, readily cut out and tried ; and again 
others cut and tried, until the shapes of good working wheels are 
arrived at, while by any other method, coupled with contempt for 
templets, there might be total failure. 

Second. Unlimited Alternate Motions. 
Mangle Wheels and Racks. 

The pitch lines of these movements have been considered in 
Figs. 102 and 103, and the forms of teeth for these pitch lines 
present no new problems. For the treatment of the various cases of 
non-circular pitch lines, and teeth for the same which are likely to 
arise in connection with this subject, we may refer to the princi- 
ples already given. 

For a mangle rack with variable velocity-ratio, the pinion for 
the same may be non-circular when the rack pitch line is to be 
curved as in Fig. 39 or Fig. 46, in order that the axis of the pinion A 
may be more nearly stationary during a movement forward or 
back. 

For a mangle wheel, the pinion may be non-circular and the 
pitch line wavy that the axis may follow a smooth or non-serrated 
groove, as in Fig. 102, or 103. 

The pitch line on the mangle wheel, besides being wavy, may be 
varied in general outline from smaller to greater radius, even mak 
ing several turns around the axis of the driven wheel B and return 
to the starting-point. In this case the velocity-ratio will vary for 
the non-circular form of pinion, A, as well as for the wandering form 
of pitch line on the wheel B. The velocity-ratio in this case will 
consist of a very complicated cycle of changes. 



CHAPTER XII. 

TEETH OF CIRCULAB GEARING. 

DIRECTIONAL RELATION CONSTANT. VELOCITY-RATIO 
CONSTANT 

FIRST: AXES PARALLEL. 

No principles in the theory of gearing, beyond those already 
treated under non-circular wheels, remain to be brought out here, 
the leading topic for our present study under circular gears being 
the special features pertaining to circular gearing, and some prac- 
tical suggestions concerning them. 

Circular gearing, though all coming under one general theory 
in common with non-circular, viz., the development of tooth curves 
by the rolling or describing templet, may yet be classified and 
named in two, and possibly three, sections, viz. : 

I. Epicycloid al Gearing. 
II. Involute Gearing. 
III. Conjugate Gearing. 

I. Epicycloidal Gearing. 

Here, not only the pitch lines are circles, but the describing 
circles also, so that the tooth outlines are epicycloidal curves / 
described by a tracing point in the circumference of a rolling circle, 
or generatrix, as the latter rolls on the periphery of the pitch circle 
as directrix. 

When the rolling is on the outside the generated curve is an 
epicycloid, and when inside it is an hypocycloid ; tooth curves here 
being therefore always epicycloids and hypocycloids. 

128 



CIRCULAR GEARING. 



129 




Fig. 147. 



SOME PECULIAR PROPERTIES OF EPICYCLOIDS AND HYPOCYCLOIDS. 

In Fig. 147 A is a pitch circle upon which the describing circle 
D rolls carrying the tracing point P and tracing the curve FPGH. 
This curve, thus generated by the 
rolling of one circle upon another, 
is called the epicycloid. If the 
rolling were inside the directing 
circle A, as in Fig. 148, it is called 
the hypocycloid. 

One peculiar property of the 
epicycloid is that it may be gen- 
erated by the rolling of two dif- 
ferent circles upon A; viz., first, 
PD rolling at I with the tracing 
point P tracing the epicycloid 
FPGH-, and, second, PE rolling 
at H with a tracing point at P, 
provided that the diameter of PE, less the diameter of PD, equals 
the diameter of A. With this relation of diameters, PD = AE, 
DA — PE, and triangles PDI, PHE, and 1HA are all similar tri- 
angles. These conditions remaining permanent during the rolling, 
P must remain on the line through /and i7as instantaneous cen- 
ters of motion, so that PI or PH remain a normal to the epicycloid 
at P from which it appears that one and the same epicycloid FPGH 
will be described by either circle, rolling as described, the difference 
of the diameters of which circles equals 
that of the directing circle A. 

Similarly, if two circles as D and E 
roll inside of the pitch line A, as in Fig. 
148, the sum of whose diameters equals 
that of the pitch line A and carry a tracing 
point P, they will generate one and the 
same hypocycloid FPG; for constantly 
PD = AE and DA = PE, and the tri- 
angles IDP, 1AH, and PEH are all mutu- 
ally similar, and hence P must describe the same hypocycloid as 
stated. 

When the two circles D and E are equal, as in Fig. 149, the 
hypocycloid becomes a straight line and is a diameter to^l, as in 
the White's parallel motion. 




130 



PRINCIPLES OF MECHANISM. 




Epicycloidal gearing is made in considerable variety, as best 
adapted to various purposes, and may be 
classified as follows: 

1 . Flanks radial. 

2. Flanks concave. 

3. Flanks convex. 

4. Interchangeable sets. 

5. "Pin gearing." 

6. Rack and pinion. 
Fig. 149. 7. Annular wheels. 

Treating these separably, we have 

1st. Flanks Radial. 

In Fig. 150, we have a pair of pitch circles, for wheels to be 
fitted with epicycloidal teeth having radial flanks, A being the 
driver, B the follower, and C 
the point of tangency. 

For convenience, we will 
take C as the origin of all the 
tooth curves. Then, as in Fig. 
112, the describing curve, in 
this case the circle D, is placed 
inside of the pitch line D with 
the tracing point a at C, and 
rolled, without slipping, to the 
left along the inside of B, the 
tracing point a describing the 
tooth curve Ga. Also the 
circle D f of the same diameter 
as D is placed on the outside 
of A with its tracing point b 
at C, and rolled toward the FlG - 15 °- 

left, the tracing point b describing the tooth curve Cb. The 
circles D and D' must be of the same diameter, or may be the 
same circle and tracing point, used in succession inside of B and 
outside of A. The tooth curve Ca is to serve as a flank of a tooth 
for B and the curve Cb for the face of a tooth of A, the two mat- 
ing and working correctly together for contacts at the right of 
ACS. (See Fig. 113 and explanations.) 

In like manner the circles, or the same circle, E and E' are 
rolled once inside of A and once outside of B, with the tracer c or 




CIRCULAR GEARLNG. 131 

d starting at C and tracing the mating tooth curves Cc and Cd, the 
former for a flank of a tooth of A and the latter for a face of a 
tooth of B, the two furnishing contacts at the left of C, all as 
explained in Figs. 114 and 115. 

We have, now, cCo a contiguous face and flank or a full tooth 
curve for one side of a tooth of the wheel A, and likewise aCd the 
one side of a tooth of the wheel B, which teeth mate perfectly, and 
will work together smoothly transmitting motion from the driver 
A to the follower B with a constant velocity-ratio of the same value 
as if the directing or pitch circles A and B were rolling one upon 
the other without slipping. 

The pitch lines being here circles, other tooth curves described 
with the same describing circles will generate tooth curves which 
will be copies of those already obtained, as cCb and aCd. Hence 
it is only necessary to copy these curves as often as needed at the 
proper pitch spaces around the wheels, when, on describing the 
addendum and dedendum circles, the drawings of the wheel teeth 
will be completed, fillets being struck in as hereafter explained. 

The leading peculiarity of these teeth is that they have radial 
flanks due to the fact explained in Fig. 149 that the describing- 
circles have diameters equal half those of the pitch circles within 
which they roll, and hence the styling radial jlanhs for these 
wheels. 

Following the contact between these teeth from beginning to 
end, for A turning right-handed driving B, we find that the initial 
contact between a pair of teeth occurs inside of the pitch circle A 
to the left of C, and moves on through the point C and ends inside 
of the pitch circle B at the right of G, and following along an " S " 
shaped curve; for full discussion of which, see path of contact, limit 
of contact, etc., Fig. 177. 

For convenience in the drawing of radial flanked teeth, we see 
from the above that it is only necessary to draw flanks as radial 
lines, and faces by using a rolling circle on the outsides of a pitch 
line, the diameter of which equals the radius of the mating pitch 
circle. 

For proportions of teeth, see Fig. 110 and rules accompanying. 



132 



PRINCIPLES OF MECHANISM. 




Fig. 151, 



2d. Flanks Concave. 
In Fig. 151 we have a pair of pitch lines, A and B, to be equipped 

with epicycloidal teeth which have con- 
cave flanks, the object of which may 
be to give to the teeth greater strength 
than if the flanks were radial. The 
increased thickening of teeth in the 
flanks may be judged by noting the 
departure of the hypocycloid from the 
radius in Fig. 151. 

The size of the describing circle is 
arbitrary, and may be chosen to meet 
the most fastidious taste or fancy as 
to thickness or strength of tooth, the 
only restriction being that the diame- 
ters of the circles D and D' be the 
same, and likewise for E and E\ 

The complete tooth curve for A,. 
and also for B, is shaded to distinguish it. A templet may be 
fitted to this curve and mounted to swing around the center, and at 
every point for a tooth curve it may be held and the tooth curve- 
marked or copied from the templet, as explained among practical 
operations. (See Page 153.) 

With the tooth curves marked off around the wheels, the ad- 
dendum and dedendum circles drawn in, and the root fillets struck, 
the tooth outlines for the wheels will be completed. 

3d. Flanks Convex. 
In Fig. 152 the rolling circles have diameters which are greater 
than the radii of the wheels within which 
they roll, as in the case of the wheel E in 
Fig. 128, or J, Fig. 119, giving flanks 
of teeth that are convex. The finished 
teeth for this case will be found very 
weak, as evident from the undercut ap- 
pearance which the teeth present. The 
filleting at the root will help this, though 
the teeth are still weaker than in the 
previous cases, but will be found of 
favorable form in light-running machin- 
ery. 




Fig. 152. 



CIRCULAR GEARING. 



133 



4th. Interchangeable Sets of Gears. 

These wheels are called interchangeable because, of any number of 
them of various sizes having the same pitch and describing circle, 
any two will work together correctly. This is seen not to be the 
case with Figs. 150 to 152. 

The essential requirement for this in- 
terchangeability is simply that the gener- 
ating circles D, D', E, and E' be always of 
the same diameter for generating faces and 
flanks of teeth throughout, as shown in 
the circles D, D, etc., in Fig. 153. 

For instance, A and B is a pair of 
wheels differing in size and selected at 
random from the set, neither being the 
largest nor smallest, for which we have 
a pair of tooth curves in contact at C, 
shaded as shown, the one aCd belonging 
to wheel A, another bCc to wheel B, etc. 

Now, the curve aCd is throughout de- 
scribed by the rolling circle D, and, as the 
latter is chosen independently of BB', 
etc., it appears that this tooth curve aCd 
is entirely free of all reference to any other 
wheel of the set, and only dependent upon Fig. 153. 

the pitch line A and describing circle D. The same is true of any 
other wheel of the set, as A', B, B\ etc., the tooth curve of any 
one wheel being peculiar to that wheel only, since the describing 
circle is one and the same throughout the set. 

Hence, to realize an interchangeable set of epicycloidal gear 
wheels, where any two whatever will mate correctly, it is only 
necessary, in addition to a constant pitch, that the describing circle, 
generating the teeth, be one and the same throughout. The 
"- change wheels " of engine lathes in machine shops is a well-known 
example of gears of this kind; that is, in "sets." 

In Fig. 153, three wheels of a set of this kind are shown with 
tooth curves drawn where the describing circle is in common for all. 

At the size B f the flanks are radial, as in Fig. 150, because here 
we strike the size where the radius of B' equals the diameter of the 
describing circle D. For wheels of this set smaller than B r the 
flanks would be convex, as in Fig. 152. 




134 PRINCIPLES OF MECHANISM. 

As the wheels of the set are made larger, that one with infinite 
radius becomes a rack, where the curves for faces are identical with 
those for flanks. 

If we carry the variation of curvature still further in the same 
direction, the rim of the wheel becomes concave, and the smaller 
wheel of the pair finds its place inside the larger. This is some- 
times called internal or annular gearing. In this case a given an- 
nular wheel has tooth curves which are exactly the same as those 
for a wheel of the same diameter with its mating gear in outside 
action. 

In the well-known example of the " change wheels " of engine 
lathes, the gears and teeth are small, and correct teeth are not so 




Fig. 154. 
important as in sets of patterns for cast-iron mill gearing with 
larger teeth, where the manufacturer must be prepared to furnish 
pairs of gears of almost any sizes with correct teeth. The best way 
to meet this demand is to have patterns made in sets, as above ex- 
plained, any two of which will mate correctly; then with all of this 
set an occasional new pattern will work correctly, one new pattern 
thus providing for many new pairs. 

5th. Pin Gearing. 

This is called pin gearing because the teeth of one wheel are pins 
and of the other are spurs to engage the pins, as shown in Fig. 154. 

To draw correct working gearing of this kind, take A and B as 
the pitch lines, and the circle at G as the section of a cylindrical pin 
or tooth. Draw an epicycloid CD, with B as the describing circle. 



CIRCULAR GEARING. 



135 



This epicycloid will be the path of the center of the pin G, on the 
supposition that the pin is made fast to the circle B, and that the 
latter is rolled, without slipping, along the pitch line A, as shown. 
The circles along the epicycloid CD represent the pin tooth in 
various positions agreeably with corresponding positions of the 
pitch line B, in its rolling, and the curve ab, just tangent to these 
circles of the pin, will be found a correct tooth curve for one side 
of a tooth or spur of the wheel A, by aid of which all the teeth of 
the wheel A may be drawn. . The pair of gear wheels A and B may 
therefore be fully drawn, the pins and teeth being properly dis- 
tributed in pitch. 

A brief examination of Fig. 155 will show that the contacts 
between these teeth and spurs are on one side of the line of centers 
AB, receding if A is driver, and approaching if B is driver. 
Also, it is found that the receding contacts 
are the most efficient, as clearly evinced by 
the extreme case of Fig. 156, where, with A 
for driver, the extended tooth curve acts 
like a wedge to force the pin along in its 
path, while if B is driver the pin is forced 
down upon the tooth curve of A, the latter 
serving very much to block or to hinder the 
movement of B; and when, as in practice, 
friction between the pin and tooth is con- 
sidered, the problem is still more prejudiced. Fig. 155. 
Fig. 15G is an extreme case to illustrate more forcibly, but in a more 
moderate one, as in Fig. 155, the principle still holds to a certain 
extent. 

As tooth contacts are on one side of the 
line of centers, the arc of contact is consider- 
ably limited, and each problem should be exam- 
ined with care to determine whether one pair 
of teeth quit contact before the next succeeding 
pair commence it, the latter usually occurring 
when the center of the pin is some considerable 
Fig. 156. distance past the line of centers. (See Mac- 

Gord's Kinematics, page 210.) For large pins, the amount to allow 
for this is greatest, and vanishes as the pin diameter becomes 
zero. 

In some examples these pin teeth have been made as rollers on 
smaller pins, with the view of reducing the friction. 





136 



PRINCIPLES OF MECHANISM. 



Pin gearing has been quite extensively used in the cheaper 
brass clocks which have flooded the country within the last forty 
years. In these the wheels are drivers, as seen above to be 
advisable, and made of sheet brass, while the pinions have pins of 
straight and smooth steel wire, held at both ends in holes in a pair 
of heads or disks of brass. 




Inside Pin Gearing. 

Pin gearing is applicable in inside gearing, the curve CD, Fig. 
154, instead of being outside would be inside, and consequently an 
hypocycloid, but the same principles apply as before. 

In the special case for inside gearing, where the pinion is half 
the size of the wheel, the pins of the 
pinions move along certain diameters of 
the wheel, according to Fig. 149, and 
sliding blocks in grooves may be intro- 
duced, with holes to receive the pins of 
the pinion as in Fig. 157. 

One of the arms of B may be cut 
away and a grooved arm of A also, and 
still the movement would work complete. 

The remaining two arms of B could 
be placed at any other angle with each 
other as 45°, 87°, etc., provided the 
corresponding two grooved arms of A 
were at half the angle intervening, 
and the action would be perfect. 

Pinion of Two Teeth. 

Approximately, rectangular teeth 
may be employed instead of cylindric, 
as shown in Fig. 158, where the longer 
sides of the pins are radial with B 
when, as shown by Fig. 149, the 
circle BEF as a templet rolling on 
the circle AF, a tracing point at E 
would describe the epicycloid KDE, 
while if rolled inside of BF it would 
describe a radial side of the tooth or 
pin E for the curve KDE to work 
against. 




Fig. 158. 



CIRCULAR GEARING. 



13 r 



A is here the pinion and has only two teeth, the least number 
possible, though it may have as many more as desired. 

The construction is a little peculiar, the teeth of B alternating 
upon opposite sides of a central disk, and for A we have two 
heart-shaped pieces, KDEGH and FJID, at such distance apart 
on a hub that the disk of B may work between. Then the con- 
tacts will alternate from side to side as the rotation proceeds. 
The part AEG will just fill the space between the two teeth E 
and G, and the wheels may work either way. 

It is hardly practical to make B driver here, since the common 
normal at E passes so near to A that, considering friction, A would 
serve as a block to hinder the driving action of B. 

In Fig. 159 we have a two-leaved 
pinion A driving B as equipped with 
cylindric pins, the construction be- 
ing the same as in Fig. 154, except 
carried to the extreme case of two 
teeth for A. 

Care should be exercised that a 
beginning contact at D is fairly 
made by the time a preceding con- 
tact ceases at E. The rolling of the 
pitch line B as describing templet 
on the pitch line A C, gives rise to 
the epicycloid dotted in at EG, 
parallel to which and at a pin radius 
EG from it is drawn the tooth 
curve for the pinion A as shown, as 
in Fig. 154. 

For inside gearing, the same pinion A may be placed inside the 
same wheel B, when the pitch line of B will fall at HI, tangent to 
A) and the pins will have contacts at H and /as shown, the center 
of B being at B' , and below B by the amount 2 AC. 

Rack and Pinion. 
For the case of the rack and pinion, the latter may have either 
the pins or the teeth, the same principles applying. 

Wheels of Least Crowding and Friction. 
A peculiar form of pin gearing is obtained by use of a describing 
circle of somewhat unusual size, as in Fig. 160, the pin having its 
side, which is within the pitch circle, a complete hypocycloid. 




Fig. 159. 



138 



PEINCIPLES OF MECHANISM. 



Here the common normal makes the least possible maximum 
obliquity with the common tangent to the pitch lines, when the 




Fig. 160. 

teeth are made just so large that the arc of contact embraces but 
one tooth contact at once; this obliquity being but about half that 
for circular pins when the contacts must all be on one and the same 
side of the line of centers, as in Fig. 154, and also only about half 
the obliquity of teeth in Fig. 150. 

These wheels of all others will have the least possible crowding 
action or tendency to force the axes from each other when made as 
just specified, so that the arc of contact but slightly exceeds the 
pitch, the friction due to crowding being therefore at a minimum, 
and the pressure to turn the driven wheel will be at a minimum 
for two reasons: first, because the driving pressure between teeth is 
nearly tangent to the driven wheel, and, second, because this last- 
named pressure is to be compounded with the least possible crowd- 
ing pressure. 

But the teeth of these wheels are unusually weak, and it can be 
recommended for only light-running machinery. 



CIRCULAR GEARING. 



139 



On account of the slight obliquity of the line of action with the 
common tangent to the pitch lines, a variation of the distance 
between the centers A and B would entail less prejudicial action 
than in Figs. 150 to 159, so that in Fig. 160 we find a suggestion 
for such wheels as used on clothes-wringers, straw-cutters, etc., 
when the distance between A and B is varied in a very large ratio* 




6th. The Rack and Pinion. 

In any of the preceding cases for Figs. 150 to 160, we obtain a 
rack and pinion by giving to one of the wheels of the pair an in- 
finite radius. 

For the case of radial flanks, as in 
Fig. 150, if we make A infinite, we ob- 
tain Fig. 161. 

Here the radial flank of A will be 
simply a perpendicular to the straight 
line A C, while the face curve OB for B 
will in this particular case be an involute 
to B, since the describing circle to roll 
inside of A, which is infinite, will be in- 
finite also; ED being one position of that circle or straight line, 
carrying the tracer D, and tracing the involute, CD, as the rolling 
advances, along CE. Thus the line DE will equal the arc CE, 
and the involute tooth face CD can readily be drawn. 

For the radial flank of B and face of A the describing circle 
GH must be half as large as B, which, with a tracer H, rolled on 
A C will in this particular case trace a cycloid, CH. 

Then we will have FCH for a full tooth curve for A, and BCD 
a full tooth curve for B; and the correct teeth can be constructed. 

As before, CD and CF will be mat- 
ing tooth curves, having a working 
contact to the right of C, while the 
mating tooth curves CB and *CH will 
have contact at the left of C. 

One peculiarity of the action be- 
tween CD and CF is that only the point 
C of the line CF will go into action 
with the whole of CD, and, as a con- 
sequence, the rack teeth would, in prac- 
tice, become unduly worn at C. 




Fig. 162. 



140 



PRINCIPLES OF MECHANISM. 



For the case of concave flanks, the describing circle G, Fig. 162, 
must have a diameter less than the radius of B to describe the 
mating tooth curves CI and CH ; and any circle, J, less than 
infinity will serve to describe the mating tooth curves CD and CF. 
The line FCHis a tooth profile for the rack AC, and the line DC1 
is a tooth profile for the pinion B. With these the teeth can be 
fully drawn for the rack and pinion. 

The curves CFand CH will be cycloids,with bases on the line A C 

For the case of flanks convex, the describing circle, as in Fig. 
152, to roll inside of B, must have a diameter greater than the 
radius of B; but for the rack, convex cycloidal flanks are impossible 
for the reason that the straight- line flank of Fig. 161 is struck with 
an infinite circle, the largest possible. Also, it is seen that by roll- 
ing any describing circle along CA, from C towards A, the line 
CF could not be described as downward toward the right, for the 
reason that no instantaneous center on CA. will serve for striking a 
curve in such position. The straight line CF, normal to CA, 
Fig. 161, is therefore the extreme case of carrying F toward the 
right. 

The Case of the Rack in Interchangeable Sets has already been 
treated in connection with Fig. 153, also Pin Gearing, in connec- 
tion with Figs. 155 and 159. 

7th. Annular Wheels. 

In Fig. 163 we have the general case, the smaller pitch line 
being within that of the annular wheel. The circle carrying the 

tracers H ov I is rolled inside of A, and 
also inside of B, to generate a face of 
a tooth of A and a flank of B. These 
curves are both hypocycloids; but, ac- 
cording to the general theory, are seen 
to be the proper ones to work together 
for a face and flank. For the outside 
curves the circle J carries the tracer 
D or F to describe the epicycloids CF 
and CD as mating curves for a face of B 
and flank of A . 

Both sets of flanks are here concave, 
and that for A must always be so; 
though that for B may be radial or convex by using a larger de- 
scribing circle to carry the tracers H and. I. 




Fig. 163. 



CIKCULAR GEARING. 



141 



A peculiar case of interference of the teeth of the annular wheel 
and its pinion occurs when the sum of the diameters of the describ- 
ing circles G and J exceeds the difference of diameters of A and 
B, as discovered by A. K. Mansfield and Prof. C. W. MacCord. 

The limit occurs when the added diameters of the describing 
circles G and J just equals the difference of diameters of the pitch 
lines A and B, as shown in Fig. 164. At this limit the faces of the 
teeth of the internal gear A have correct acting tooth contacts with, 
the faces of the teeth of the in- 
side pinion B, as may be proved 
by aid of the alternate describ- 
ing circle K. 

Referring to Fig. 148, it is seen 
that the hypocycloid DL, Fig. 
164, may be regarded as being now 
traced by either G or K carry- 
ing the tracer D; and that aDC 
is a straight line. Also referring 
to Fig. 147, it is clear that the 
epicycloid DM may be regarded 
as in the act of being traced by 
either J or K carrying the tracer 
D, and that the points DbCuve in a straight line. Hence, aDbCare 
points in one and the same straight line; and the hypocycloid DL 
and epicycloid DM are tangent to each other at D, as required 
for tooth contacts; and we have, at this limit, in addition to the 
usual face and flank contacts, this face to face contact at D. 

If D -J- J is larger there will be interference at D between faces, 
while if smaller this face to face contact is lost. 

In this particular case we have the advantage over ordinary 
gears of an additional and unusual contact at D, and a most excel- 
lent one, except for its remoteness from C, where the amount of 
slip of face surfaces over each other is greater than in the other 
and usual contacts. 

From what has been said, it is plain that the describing circles 
G and J can be varied, either smaller than the other, but that they 
must not be larger than that they will fit in between the two pitch 
circles, as shown, under penalty of positive interference of the teeth 
at D, but that they may be smaller than shown, with loss of con- 
tacts at D. 

Wlieels in sets may be worked in annular gearing, that is, any 




142 PRINCIPLES OF MECHANISM. 

of a number of pinions may work correctly in any of a number of 
annular wheels, provided that one and the same describing circle 
be used throughout, and that the facts of interference be observed, 
as pointed out above. 

Pin annular gearing may also be used by following the princi- 
ples explained under pin gearing, and the pins may be in either the 
wheel or pinion. 

II. Involute Gearing. 

As explained in connection with Figs. 119 and 121, logarith.nic 
spirals may be employed for describing curves which, rolled on the 
pitch lines A and B, inside and outside in the usual way, will de- 
velop faces and flanks by pairs, the curves thus traced being invo- 
lutes to circles somewhat smaller than the pitch lines. 

Thus, in Fig. 165, take the pitch line A — DCH and roll the 
logarithmic spiral FD from C as shown, with a 
tracer at the pole F. The curve CF will be an 
involute to a circle EG, inside of A, to which the 
line EDF is a tangent. This is made evident 
by the fact that the logarithmic spiral is a carve 
of constant obliquity, that is, that any line or 
radius vector, FD, makes a constant angle with 
a tangent to the spiral at D, and, consequently, 
Fig. 165. a cons tant angle with the tangent to the circle at 

D, and also with a radius to the circle at D. Hence, FDE will al- 
ways be tangent to some one circle EGA) and it is plain that we 
will obtain one and the same curve, CF, whether the latter be gen- 
erated by the rolling of the spiral, with tracer F, along CD, or by 
the unwinding of the cord FE, with tracer F, from the circle GE, 
This curve, CF, will thus be an involute to the circle A GE. This 
curve as an involute starts at G, while as developed by FD it 
seems to start at C. To supply CG by the rolling spiral, place the 
pole and tracer at C, and roll inside along CH as shown, when CG 
will be traced just to G and stop; for then the radius of curvature 
of the spiral at H will be HA, since to construct this radius of 
curvature for a logarithmic spiral draw the right-angled triangle 
A GH, when AH is that radius. The spiral then cannot roll beyond 
H, and the curve CG stops at G. 

As tooth curves, CF is to serve as a face and CG as a flank, and, 
as developed by the spiral, are seen to fall within the general theory 
of development by rolling curves. But, in practice, for convenience, 




CIRCULAR GEARING. 



143 



the curves are drawn as involutes to the circle GE, called the base 
circle or base of the involutes. 

When the teeth need to be cut deeper between than CG, such 
portion is cut away arbitrarily below EG as shall make room for 
the interengaging teeth. 

To Obtain Tooth Curves for Wheels with Involute Teeth. — Draw 
the pitch circles A and B in common tangency at C, Fig. 166, 
then a straight line D CE through the point of common tangency 
or pitch point C, and perpendic- 
ulars AD and BE from the cen- 
ters A and B upon this line. 
Then, with AD and BE as radii, 
draw the base circles FD and HE 
of the involutes. The involute 
may be drawn by unwinding a 
thread from the base circle, the 
thread having the tracer attached, 
or by drawing several tangent 
lines as shown, and stepping with 
dividers from C to D, then with 
the same number of steps to F, 
then back to other points of 
tangency and out on the tangent 
to points G, etc., for as many 
points as desired, when the invo- 
lute FCG maybe drawn in; likewise for HCI. 

This gives us a pair of tooth curves from which the teeth may 
be drawn in, clearance room within the base circles DF and EH 
being assumed. As the finished gear wheels revolve, the tooth 
contacts all occur on the line DE, remaining on and moving 
along it; and there can be no correct contacts outside of the 
length DE, though between D and E there may be several. At- 
tempted contacts beyond D or E give rise to serious interference. 

In practice it is rare that the contacts extend to the whole of 
DE, though they may, by laying out the wheel with that result in 
view, as by making the addendum circles cut at D and E. 

These gear wheels possess the peculiar property that the dis- 
tance between centers A and B may be varied at pleasure, without 
altering the correctness of action of the teeth upon each other. 

For, examining Fig. 166, we find that the triangles ACD and 
BCE are similar, so that the pitch point C is always at the same 




Fig 166. 



144 



PRINCIPLES OF MECHANISM. 



proportional position between A and B, whatever the magnitude 
of AB, allowing AD and BE to remain constant. Under these 
conditions any number of correct working tooth outlines may be 
drawn with varied distances AB. The spaces between the teeth 
along on the line FD, or on DE, or on HE all equal the normal 
pitch, which is constant; while the spaces between the teeth along 
on the pitch circles equal the circumferential pitch. This latter dif- 
fers slightly with the normal pitch, and is slightly variable if we 
regard the pitch circles as remaining tangent to each other at C 
while the line of centers AB varies. 

This gearing is often recommended for situations where the 
distance AB, through carelessness, wear of parts or otherwise is 
subject to changes ; and where epicycloidal gearing, requiring AB 
to remain constant, will not serve correctly. 

These involute wheels in a group of various sizes of the same 
normal pitch will all work together interchangeably, and they are 
probably the best for change gears for engine lathes and the like. 
In the Rack and Pinion the wheel A may be regarded as of in- 
finite diameter when the pitch line becomes 
the straight line AC, Fig. 167. The base 
circle also becomes infinite and at an infinite 
distance from C, so that the line CD is infi- 
nite and the tooth curve FG for the rack 
becomes a straight line perpendicular to 
DE, the face and flank being both parts of 
one and the same straight line. The tooth 
Fig. 167. curve, HI, remains the same as before, for a 

like inclination of DE. The addendum line for the rack should 
never go above E. 

For the Annular Wheel and Pinion, the tooth curves for the 
former become concave as at FG, Fig. 168, 
while the pinion remains the same as be- 
fore for the same inclination of CED; and 
for this a group of pinions of the same 
pitch will interchange with this wheel. 

In the present case there can be no 
contacts between D and E, but may ex- 
tend from E through C to infinity, except 
for interference at L. 

In these gears, as well as in epicy- Fig- 168. 

cloidal, there may be interference of teeth, as shown in an exaggerate 





CIRCULAR GEARING. 



145 



ed view at EL and KM, the tooth curves intersecting at K. This is 
most likely to occur where the pinion is relatively large. To examine 
a case for interference, draw the pitch circles and on them step off 
equal pitch distances from C to M and JY, and through these 
points draw the involute tooth curves, as shown. If these curves 
do not intersect, as at K, within the addendum of the pinion and 
the circle DL of the wheel, there will be 
no interference at that position, and, if 
the test is applied at the intersection of 
the addendum circle for B with the base 
circle to AD, with no intersection of 
tooth curves, there will be no inter- 
ference of teeth for the wheels. Modifi- 
cations of amount of interference are 
affected by varying the addenda and the 
inclination of CED. 

Different systems and sizes of teeth 
may he combined in a single gear, as 
illustrated in Fig. 169 of a pair of cir- 
cular gears, 2 to 1, velocity-ratio constant, embracing both the 
epicycloidal and the involute forms of teeth. 




Fig. 169. 



III. Conjugate Gearing. 

This term is applied to teeth where the tooth of one wheel is 

assumed and the mating tooth for 
the other wheel found from it, 
usually by the practical operation 
of templets, as foreshadowed in 
Fig. 108 and more definitely shown 
in Fig. 122, which may be followed 
fully here with circular pitch lines. 

SOME PARTICULAR CASES. 

1st. Flanks Parallel Straight, 
Lines. 
In Fig. 170, A and B are the> 
pitch circles somewhat removed 
from tangency to better show the 
work. One of the parallel flanks 
is CG. From step off equal 
spaces, marked 1, 2, 3, 4, etc., or, 




146 



PRINCIPLES OF MECHANISM. 



the pitch line of B. Likewise from C step off the same equal spaces 
on the pitch line A as shown, C and C being a half-tooth .thick- 
ness from the line of centers. Draw la, 2b, 3c, etc., perpendicular 
to CG. Then with la, from wheel 5asa radius, draw a circle arc 
about the point 1 on the wheel A ; also, with b2 as a radius, draw 
a circle at 2 on A. Likewise with bd, etc., for as many points as 
required. Then draw an enveloping curve FC f , which will be 
a correct face curve for A, to work on the flank CG as mating 
tooth curves. Other pairs of mating face and flank lines may be 
drawn, and the teeth completed, as shown in dotted lines. 

Examining the mating curves drawn, it is plain, for example, 
that the radius from 3 to where its circle is tangent to the envelope 
C'F is a normal to G'F, so that when the points 3 of pitch line A 
and B are tangent to each other this normal will coincide with the 
normal 3c, and c will be the point of contact and tangency between 
the tooth lines. Likewise for other points, so that the face and 
flank will slide smoothly with each other throughout, while the 
pitch lines mutually roll, thus answering to required conditions for 
tooth action. 

2d. Flanks Convergent Straight Lines. 

In Fig. 171 draw the pitch 
lines and the convergent flanks 
CG. Step the equal spaces 1, 2, 
3, 4, etc., draw the lines al, b2, 
etc., normal to the flank line 
CG, use these latter as radii, and 
draw circle arcs as before on the 
pitch line A. The envelope C'F 
will be the proper tooth face for 
the wheel A, to work on on the 
flank CG, and the complete 
tooth can be drawn as in Fig. 
170. 



QB 




Fig. 171. 



3d. Flanks Circle Arcs. 

Draw the pitch circles A and B, the line of centers, and a tan- 
gent HI at H, Fig. 172. From a point /on HI draw the circle arc 
CG for the flank of a tooth for B. Through equidistant points 1, 2, 
3, etc., draw radii from I. Then al, b2, etc., are the radii to use 
at points 1, 2, 3, etc., on A as centers of arcs, tangent to which 



CIRCULAR GEARING. 



147 



tooth of A 
QB 



that will 



draw the envelope C'F for a face of a 
work correctly on the flank CG. 

As another example, the cen- 
ter 7 may be taken anywhere be- 
low the line HI, but no higher 
above than to be on a tangent 
to the pitch circle B at the point 
C, in which case CG will be nor- 
mal to the pitch line. No example 
is to be found of a correct flank 
line which is undercut as by 
placing I still higher. 

Again, in another example, 
the line CG may be a parabola, 
ellipse, or other curve to suit the fancy, always being so drawn as 
not to undercut at C, and in every case the lines a\, b2, etc., 
being drawn normal to the curve CG. 

The inverse of these cases is practicable where the face of a tooth 
of A is assumed to be a circle, parabola, or other curve, and the 
flank found to match it in a similar way. 




CHAPTER XIII. 

PRACTICAL CONSIDERATIONS. 

So far in circular gearing, we have dwelt upon the theory of 
tooth carves which are perfect in action. But the demands of 
practice are not always so severe as to require the teeth to be thus 
accurate. There are other considerations also to be noted, such as 
proper addenda, clearance, fillets, obliquity of action, overlap of 
action, practical methods, approximate teeth, etc. 

ADDENDA AND CLEARANCE. 

At Fig. 110 is given a set of rules for proportioning the teeth, 
which are -among the simplest of quite a variety. In cast gearing 
more clearance is required than in machine-dressed teeth, and more as 
cast at some foundries than at others, according to care in moulding. 

The rule for circumferential 
pitch at Fig. 110 will answer for 
the average of care in moulding,, 
and for ordinary sizes. For 
very large teeth the propor- 
tion for clearance is too great, 
while for very small cast teeth 
Fig- 173. it will often be found too small, 

so that the best rule will probably be the one stated varied to suit 
the judgment of the draftsman in each particular case. 

The rule as to diametral pitch is best adapted for cut or machine 
dressed gear teeth, where a comparatively slight bottom clearance 
is sufficient and a much less side clearance. The Brown & Sharpe 
cutters and instructions are apropos here. 

148 




PRACTICAL CONSIDERATIONS. 



149 



In Fig. 173, for cast teeth, cd is the pitch circle, e the addendum 
circle, / the dedendum or root circle; c being equal to 5/11, d 
equal to 6/11, a equal to 3/10, and b equal to 4/10 the pitch, ac- 
cording to the rule of Prof. Willis, as given at Fig. 110. 

To Strengthen the Teeth, root fillets are generally struck in as at 
jk, which only extend to a part of the bottom surface at k and up to 
a comparatively small portion of the flank at/. In most cases the 
bottom of the space may be limited by a circle arc, ghi, tangent to 
the flank lines at g and i, and to the root circle, thus simplifying 
the shape of the bottom of the space and greatly strengthening the 
teeth. To test this matter the so-called " clearing curve" may be 
drawn — a curve that would be described by the corner of the point 
of the tooth, as in Fig. 174. 

To Find the Possible Clearing Curve described by the tooth 
corner b, Fig. 174, as it enters the space clef, the side ab acting against 
the side de, make equal spaces 1, 2, 3, etc., 
then with bl for a radius describe a circle 
arc from the point 1 near d; again, with b2 
for a radius describe the arc from 2 in the 
space near d, etc., for as many points and 
arcs as required. Then the enveloping 
curve to these arcs, as shown, is the possi- 
ble clearing curve. This curve, of course, 
extends only the addendum distance be- 
low the pitch line. Now a circle arc, 
drawn in as ghi, Fig. 173, where a = 3/10 
and b = 4/10, will usually, if not always, Fig. 174. 

be found to go outside of this possible clearing curve, and, as it is 
the simplest form of outline for the bottom of the space as well as 
gives greater strength of tooth than does the smaller fillets jk, 
Fig. 173, it is recommended. Indeed, if ever found necessary, it is 
usually advisable to go a little deeper to get in the circle arc rather 
than adopt two smaller arcs. The circle arc is dotted in, and is seen 
to go considerably below the curve enveloping the circles struck 
from 2, 3, etc., Fig. 174. 




THE PATH OF CONTACT. 



The Path of Contact between a pair of operating teeth of mated 
gear wheels is the line followed by the touching point between the 
pair of teeth, and differs in form in different systems of gearing- 



150 



PRINCIPLES OF MECHANISM. 




Fig. 175. 



In Epicycloidal Teeth it is the Describing Circle Itself, in a cen- 
tral position, or that of common tangency 
with both pitch circles, as at C, Fig. 175 r 
where CF is the describing circle. This may 
be shown by supposing CD equal to CE y and 
that the describing circle has rolled from 
where the tracing point F was at D over to 
the position shown, and likewise from E 
over to the same position. The rolled curves 
DF and EF will be tangent to each other at 
F, because CF is a normal to both curves, since C is an instanta- 
neous axis of motion of the tracing point. Therefore F, the point of 
contact of the tooth curves DF and EF is on the describing circle. 
Hence the point of contact F follows along on the describing circle 
CF from C to F, the tooth curves engaging at C and move toward 
the position shown, making the describing circle CF the path of 
contact. 

In Involute Gearing, it is plainly seen from Fig. 166 that the 
path of contact is on the line DCE. 

In Other Cases, the Path of Contact may Eeadily be Found. To 
illustrate, let us find it for the teeth 
of Fig. 172 as shown in Fig. 176. 
Draw radii from B to the points 1, 2, 
3, where normals to the flank curve 
Cc cut the pitch circle. Draw circle 
arcs through the points a, b, c, where 
the normals meet the flank curve, and 
extend them to meet the radii from B. 
Then, evidently, the flank curve is 
in contact at a, with its mating face, Fig. 176. 

when the point 1 is at H on the line of centers, and a at % where- 
im — ae; thus making i one point in the path of contact. Like- 
wise, jn == bf, ko = eg, etc. ; thus giving the curve Hijh as the 
path of contact for the pair of tooth curves of Fig. 172 as above 
the pitch lines. Usually the path extends below the point of com- 
mon tangency of the pitch lines toward A, for the mating tooth 
curves below the pitch lines as well as above toward B, and this 
path is always a curve, except in involute teeth. 

By the Inverse of this the Path of Contact may be Assumed, and 
the tooth curves, face and flank, may be found therefrom, as was. 





PRACTICAL CONSIDERATIONS. 151 

done by Prof. Edward Sang, and explained in his remarkable treat- 
ment of the subject of gear teeth published some years ago. 

The Path of Contact is Limited in Practice where the addendum, 
circles cut the path lines. Thus, in Fig. 
177, draw the addendum circles EF and 
DF. Then the contacts for the epicycloi- 
dal teeth of the wheels A and B must 
begin and end on the S-shaped line or 
limited path DCE. One pair of teeth 
should engage at D as soon as the preced- 
ing pair quit contact at E, but usually it 
is sooner. In involute teeth, the contact 
cannot possibly extend beyond the points Fig. 177. 

D and E, Fig. 166, so that the addendum circles should not over- 
reach those points as has been stated. 

The Acting Part of the Flank of a tooth, that is, the entire 
portion that engages with its mating face, is simply that which ex- 
tends from the point D to the pitch line of B, or from E to the pitch 
circle of A, Fig. 177. This portion of the flank is much shorter, 
usually, than its mating face, thus giving cause for more rapid 
wear on the flank than face, from which it appears that gear teeth 
will wear out of shape and not into it, so that if teeth are ever 
correct in form, they must be made so. 

The Line of Action of the Pressures of Contact between the 
teeth, neglecting friction, is always in the line of the normal, as 
OF, Fig. 175; this line being therefore properly called the line of 
action. In involute gearing it is always one and the same straight 
line, DCE, Fig. 166, but in all other forms of teeth it is a line of 
varying inclination, like a straight line, iH, jH, JcH, etc., Fig. 176. 

The tendency of the teeth to crowd the wheels apart depends 
upon an intermediate obliquity of this line of action, the same 
being reduced as the maximum obliquity is reduced for a given sys- 
tem of teeth. 

Thus in epicycloidal gears, the larger the describing circles, CD 
and CE, Fig. 177, the less the obliquity and the tendency to crowd.. 
The maximum obliquity of the line of action with the common 
tangent to the pitch lines is that of the straight line through Z> 
and C on the one side, and the line EC on the other. 



152 



PRINCIPLES OF MECHANISM. 



The Blocking Tendency. 

The tendency of a pair of teeth to " block " or stop the wheels 
is greater in approaching action than receding, and increases with 
the obliquity of action; so that in wheels of few teeth the maximum 
-obliquity of the line of action should be kept as small as possible, 
.as already explained under non-circular gearing, and sometimes it 
may be advisable to make the addenda unequal to favor the obli- 
quity in approaching action. 

Unsymmetrical Teeth. 

These maybe made when desired, by using smaller describing 
•circles for one side of a tooth than for the other, for epicycloidal 
teeth, or greater obliquity of the line of action in involute, etc. 
J?ig. 169 illustrates several such teeth. 

To Draw the Tooth Curves in Practice. 

Epicycloidal Tooth Curves may be drawn as in Fig. 178, where 

A and B represent a pair of pitch 
circles for which it is desired to 
draw a pair of tooth curves of this 
form for teeth with concave flanks, 
the work all being completed on 
the drawing-board and with only 
the ordinary drawing instruments. 
Having the pitch circles drawn at 
a distance apart, if need be, for per- 
spicuity, adopt some diameter of 
describing circle CH = C'H, such 
as has a diameter less than the 
radius C'B as for concave flanks 
for B. Then, with the dividers, 
describe the circle in several posi- 
tions on A, and in B, tangent to 
the pitch circles as shown. Then 
with C as the origin of a tooth curve, step off with the spacing di- 
viders on A, as at abc, etc., to near the point of tangency with A of 
some drawn circle F, say the first one, and then, without lifting 
the dividers from the paper, step backwards on this circle an equal 
number of steps to F, and note the exact point. Then beginning 
at C again, step along on A to near the tangency with the next 




PRACTICAL CONSIDERATIONS. 



153 



circle, and then back on that circle an equal number of steps to G, 
and carefully note that point. So proceed until the desired num- 
ber and frequency of noted points is obtained. Then the epi- 
cycloid, CFG, may be drawn in through the points" noted by aid of 
an irregular curve. 

In stepping from C along on A it is not necessary to come out 
exactly at the point of tangency with the circle F, G, etc., but the 
nearest half-step or less from it, and then turn back on that circle 
to the point F or G, as the case may be. Neither is it necessary, as 
F is noted, to go back to C, but without lifting the dividers we may 
retrace the steps from F to A and thence along A to near the next 
point of tangency and then backwards on that next circle, to G, etc. 

Proceed likewise for all the tooth curves to be drawn, as, for in- 
stance, the mating flank for C, F, G, using the same describing 
circle, resulting in the points C ', D, E, etc., in B. Also a face for B 
in the points L, M, N, etc., and its mating flank/, J, i$T in A, using 
the same describing circle for both. We then have a complete tooth 
profile, GFC1J, for A, and another, JVLCDF, for B. Templets may 
be formed to these curves and used to finish drawing all the teeth 
of the gear wheel A or B, either on the drawing-board or on the 




Fig. 179. 
wooden pattern being made for use in casting. When the flank is 
ready for it prepare a templet of cardboard, thin piece of wood, or 
zinc plate. The latter may be blackened with a chemical as an ex- 
cellent preparation for visibility of lines drawn upon it. Lay off 
the curve GCK, for instance, with accuracy. Also, if desired, the 
addendum circle at P, Fig. 179, and the root circle at 0, to fit the 
root curve of Fig. 173. At R cut an opening with a sharp point R 
coming exactly to one of the pitch points T, R, V, so that CR 



154 PRINCIPLES OF MECHANISM. 

equals a half -tooth thickness. Also at S cut the templet to a point 
that shall just fit upon the pitch circle. This gives us the required 
tooth templet. Then, when this templet OCPRS is placed at any 
pitch point, as R, the edge of the templet PCO is in just the posi- 
tion for drawing a scriber along the edge of the templet to trace 
the whole tooth profile, including addendum and root curve. Then 
the templet is to be shifted along the pitch line till the point of the 
templet at R just coincides with the next pitch point T, and the 
point S with the pitch line, when the next tooth profile may be 
traced, as dotted at QU. Proceed likewise around the wheel-blank, 
or the drawing on the drawing-board. Then the templet, being 
made of thin material, may be turned over and the other sides of 
the teeth all drawn, and the outlines of the teeth completed, as 
shown for the one OQU. 

A Radius Rod, if preferred, may be used instead of the points 
R and S, the templet being mounted as at X on the rod AX by 
tacks or screws, and a center pin struck through the radius rod, and 
at the exact center point A, by means of which the templet may be 
swung around the wheel and always remaining in the exact tooth- 
curve position ; it being only necessary to stop and hold it at each of 
the several pitch points, while tracing the tooth curve for the same. 
The templet may be turned over on the radius rod and the other 
sides of the teeth drawn. 

For Involute Teeth, as in Fig. 180, let A and B represent a pair 
of pitch lines for which it is desired to draw this form of tooth. 
Draw a line DOE through the pitch point C and at an angle of 
about 75 degrees with the line of centers AB. Also draw AD and 
BE from the centers A and B perpendicular to BE, and circles 
tangent to BE as shown. To the last-named circles draw a series 
of tangents JL, GM, KN, 10, as many as required. Then with 
spacing dividers step from C along on CD to near the tangency D, 
then backward on the circle an equal number of steps to F, and note 
that point. Without lifting the dividers, step back past D to near 
L and out on LJ to J an equal number of steps and note the point 
J. Then back on JL and on to near M and out on MG to G and note 
that point. So proceed for as many lines, JL, GM, etc., as the de- 
sired frequency of the points F, C, J, G calls for. Then trace the 
curve through these points, which curve will be an involute to the 
circle CDM, exactly in theory, and very nearly so by this process of 
drawing, and to serve for a tooth curve for wheel A. Likewise the 
involute HCKI is to be drawn to serve for a tooth curve for B. 



PRACTICAL CONSIDERATIONS. 



155 



Then the addendum circles and the root curves are to be drawn in,, 
giving the full tooth profiles, corresponding to PCO of Fig. 179,. 
when templets may be made and applied as explained at Fig. 179. 
The line D CE was said to be drawn at about 75 degrees with AB y 
but this angle is arbitrary, some preferring it at more and some less 
than 75 degrees. Probably the best criterion to follow as to this, 




Fig. 180. 

angle is the judgment of the designer as to form of teeth a partic- 
ular angle gives, except that the addendum circle for A should not 
reach beyond the point E, nor the addendum circle for B reach: 
beyond D, on the penalty of interference of teeth; and, at the same 
time, the normal pitch of the teeth should not exceed the portion 
of DE as intercepted between the addendum circles. The normal 
pitch is the distance from one tooth to the corresponding point on 
the next, as measured along on the line DE, this line being normal 
to all tooth curves that intersect it between D and E, 



Approximate Teeth in Practice. 

In many cases, a simple circle arc will answer the requirement 
for a face curve and also flank, especially in gear patterns where the 
inaccuracy of moulding of the gears in moulding sand enters the 
account; and for moderate sizes. But it is easy to select a curve 
that will approximate the epicycloid or the involute closer than 
the circle will, as, for instance, an ellipse, though a series of ellipses 
of various sizes would be needed to select from in a particular caso 
for the best results. To avoid this, a parabola or hyperbola may, in 
a single curve, meet all sizes of teeth, as well as a series of ellipses, 



156 



PRINCIPLES OF MECHANISM. 




Fig. 181, 



but how to locate the curve to best approximate the true tooth 
without drawing it would be a serious question with a parabola or 
hyperbola. Further thought suggests a spiral as having advan- 
tages, and, finally, the logarithmic spiral, by reason of its peculiar 
and simple properties, is perceived to be the best adapted of all 
curves for this purpose. Thus it is well known that a normal to 
this spiral at any point, a, will be tangent to an equal logarithmic 

spiral EcF, Fig. 181; 
and that ac will equal 
the length of the curve 
cE, and also equal the 
radius of curvature of the 
logarithmic spiral EaG 
at a. This shows that 
the two equal copolar 
logarithmic spirals EaG 
and EcF are mutual in- 
volute and evolute curves. 
Also, it is known that 
the triangle Eac is right- 
angled at E, 
It is found by trial on a carefully made drawing that, taking any 
point d as a center on ac produced, and drawing a circle through 
a as dotted, the circle will cut the spiral at such point, b, that the 
length cd, divided by the length be, will be very nearly a constant, 
at least when the obliquity of the spiral is such that Ec equals 2Ea, 
as in the case of the logarithmic spiral, which is found to best fit 
the epicycloidal tooth curve. 

Now draw a pitch line IJH for a gear wheel and tangent to ad 
at J, so that aj will equal the half-tooth thickness. Then the 
epicycloidal tooth curve, suitable for a tooth, so drawn as to pass 
through a, will also pass very nearly through b, when the latter is 
taken at the addendum circle; due regard being paid to the pitch 
diameters and tooth numbers in drawing the epicycloid. 

The Template Odontograph has been formed of this spiral with 
Ec = 2Ea, or with the tangent of obliquity equal to % and is 
therefore a curve which very closely approximates the epicycloidal 
tooth curve. It forms at once a universal and ready-made tooth 
templet for the draftsman, which has been generally accepted as 
being a good substitute for the true tooth curve. The instrument 
is accompanied by tables for use in setting it on the pitch line in 



PRACTICAL CONSIDERATIONS. 



15' 




Fig. 182. — Odontograph set at 
setting number 2.50 on line 
AC. 



the right position for drawing the various kinds of teeth, such as; 
flanks radial, flanks concave, interchangeable sets, involute teeth,. 
etc.* 

The instrument, full size, is 
about four by six inches and is 
shown in Fig. 182. The curved 
edge at A is graduated and num- 
bered for a length of 3 inches. To 
set the instrument in position, 
" settings " are made out from 
accompanying tables, as, for ex- 
ample, 2.50. A line ABO is 
drawn, when the 2.50 is brought 
to the line at A, while the curved 
edge at B is brought just tangent to the same line ABC. 

To set the instrument for drawing the tooth of Fig. 183, draw 
the tangent AHCE to the pitch line at C, the middle point of the 

tooth, also a tangent BD at 
/ ' K H, the side of a tooth. Then,, 

with "settings" made out 
from the tables, place the 
instrument on the tangent 
HCE as explained, and, 
while retaining it there, 
draw the scriber along the 
Fig. 183.— How to place the Odontograph. edge from D outward, giving* 
the face curve. Similarly with the instrument set on BD as shown, 
draw the scriber from D inward, giving the flank curve. The 
addendum circle and root curves may then be drawn in as pre- 
viously explained, when the full tooth profile is completed. 

This Templet Odontograph, by means of screw-holes shown,, 
may be mounted on a radius rod and swung around a center pin,, 
for marking all the tooth curves, as explained at Fig. 179. W& 
thus have, in this instrument, a convenient ready-made tooth tem- 
plet for all cases. 

The accuracy of results, as compared with those of other ap- 
proximate methods, is discussed, and examples are given in the 

* For a full description of the instrument, formulas for calculating "set- 
tings," tables of settings, etc., see Van Nostrand's Science Series, No. 24, and 
Van Nostrand's Engineering Magazine for 1876 ; also a pamphle accompany- 
ing the instrument, all found in the instrument stores. 




158 



PRINCIPLES OF MECHANISM. 



articles cited in reference. Involute teeth, as well as the various 
cases of rack-and -pinion annular wheels, etc., are drawn by the 
instrument. 

The Willis Odontograph is a well-known instrument for locat- 
ing the centers of circle arc tooth curves in such a way that ?, face 
is of one circle-arc and the flank of another, for teeth approx- 
imating those of the epicycloidal form in interchangeable sets, 
where a wheel of 12 teeth has radial flanks. This instrument is 
of special interest, as being the first odontograph: originated some 
sixty years ago by Prof. Eobert Willis. 

The instrument is founded on principles made clear by Fig. 
184, where A C and BC are the radii of the 
pitch circles, CDH the generating circle 
to roll on A, to describe a face of a tooth 
of A and to roll inside of B to describe 
the flank for B, that is to mate with the 
face of A. Take the center of CDH on 
the line AB. 

Assume a point D on the describing 
circle CDH, and draw DCF, DH } CI 
parallel to DH, BIF, and A EI. Then CI 
and DH will be perpendicular to DCF, 
because CDH is in a semicircle; and E 
-will be the center of curvature at D of an epicycloid drawn through 
D by a tracer at D in the circle CDH used as the describing circle, 
as the latter is rolled on the pitch circle A. Also, i^will be the 
center of curvature at D of a hypocycloid drawn through D by a 
tracer at D, in the describing circle, as the latter rolls inside the 
pitch circle B.* 

*A proof of this is given in Van Nostrand's Engineering Magazine for 
1878, Vol. XIX, page 313, viz.: 




CD = HI 



GE = 



CD X AC 



and BE 



CE : : AH : AC 

CD X AC 



m + gp^^ + A^ 



AH . * — ' AH 

which is the expression for the radius of curvature at D for an epicycloid 
passing through D as generated by rolling the describing circle CDH on the 
pitch line A. 

In a similar way, the radius of curvature at D of the hypocycloid passing 
through D, as generated by rolling the describing circle CDH on the pitch 
line B, is 

df •= gd t 1 + ■!§)• 

See Davies and Peck's Mathematical Dictionary, page 222, for formulas for 
radii of curvature. 



PRACTICAL CONSIDERATIONS. 159 

Now, as E and F are centers of curvature of the mating epi- 
cycloid and hypocycloid tooth curves, it is plain that, by taking the 
points E and F as centers and describing circle arcs through D, 
these arcs will approximate the epicycloidal curves and serve as 
mating tooth curves with corresponding approximation to accuracy. 

Without reference to analysis, regard ICF as a triangular tem- 
plet, swinging about a pin at I. For a movement corresponding to 
that of the action of face and flank, the edge CF will not depart 
far from intersection of AB at C. This edge line, CF, would have 
nearly the same movement if, instead of one side of a triangle, it 
were jointed at E to a rod AE, since E is on the straight line AI, 
and would, in either case, move in a circle arc normal to A I. Also, 
for a like limited movement, if F of the line CF were jointed to a 
Tod BF, the point F would move in nearly the same path, so that 
motion transmitted from A to B through the medium of rods AE, 
EF, and FB, jointed at E and B, would maintain nearly constant 
velocity-ratio for the limited movement considered, as required for 
the tooth curves. 

If now, with E and i^as centers, circles be struck through D 
for a pair of tooth curves for gears A and B, their velocity-ratio 
would be the same as that for the rods connecting the axes A and 
B, because the distance between the centers E and F would re- 
main constant in either case. This last consideration holds for 
whatever point near D in the circle CDH selected, as that through 
which the circle-arc tooth curves be struck. 

Hence, to draw a pair of approximately correct tooth curves as 
circle arcs, for the pitch lines A and B, Fig. 184, draw the lines 
AB, CF produced, CI perpendicular to CF, AI, and BIF. Then, 
with E and F as centers, draw a pair of circle arcs through some 
point on CF produced. Thus, Fig. 184 is general, but for con- 
venience of millwrights Prof. Willis limits the diagram, the point 
through which the teeth are drawn being taken a half-pitch from 
C toward D, and may sometimes be the point D, as when the line 
CF is drawn, as Prof. Willis takes it, at an angle of 75 degrees with 
AB. 

To obtain a convenient odontography Prof. Willis assumes the 
points / and D to be on a circle CD HI of constant diameter for a 
given pitch and to equal the radius of a wheel of twelve teeth, for 
which D will be a half-pitch from C, and the line DCF at an angle 
of 75 degrees. This makes all the tooth-circle arcs pass through 
the point D when BE, the radius of a face, will be the radius of 



160 



PEI2STCIPLES OF MECHANISM. 



curvature at D of an epicycloid through D generated by rolling the 
circle CDH on the pitch circle A ; and DF the radius of curvature 
of the hypocycloid in B through D, thus giving for the tooth curves 
the osculating circles of the epicycloid and hypocycloid at D. A 
careful ly-made drawing will show that the point D for the above 
limit will be a little above the half-height of the tooth face; while 
it is probable that a preferable result is obtained if it be a little be- 
low, as for the case where a 15 -toothed wheel has radial flanks with 
CD a half-pitch and the line CF at an angle of 78 degrees with AB, 
TJie Willis Odontography shown at agF 9 Fig. 185, locates the 



^"'M 1 [ ' ' ' [ ' ' ' ^> >1 




centers E and i^of 'Fig. 184 with only the brief diagram of Fig. 
185, where CC is one pitch in the pitch circle, D the middle or half 
pitch point, AC and AC radii, and DL and DK a face and flank 
curve or circle arc respectively, drawn to the centers E and F. 
These centers are found by placing the odontograph at C'E and at 
CF, with the leg Ca on the radius, and noting the points E and F 
by aid of a table from which values, or distances C'E and CF are 
taken, according to radius of wheel being drawn, and noted in the 
scale gCF. When the tooth profile KDL is thus drawn, a profile 
templet may be formed as explained at Fig. 179, and the tooth out- 
lines drawn for the complete wheel. 

This odontograph is made in metal or cardboard, the latter being 
larger and having the necessary tables printed thereon. 

A simple odontograph was brought out by Prof. Willis, giving 
centers of tooth-circle arcs, when the whole profile is formed of 
one arc, thus approximating involute teeth. This instrument sim- 
ply gives the centers D and E, Fig. 166, and is of so little help in 
finding D and E as to be in slight demand. 

The Three-Point Odontograph of Geo. B. Grant is so called be- 



PRACTICAL CONSIDERATIONS. 161 

cause its application results in a circle arc struck through three points 
Cab, Fig. 186, in the actual epicycloidal face curve CD; one point, 
C, being at the pitch line ICU, a second at the middle of the face 
at a 9 and the third at b, where the addendum circle intersects the 



b/ D 



epicycloid. The position of the center E, of a circle which will pass 
through the three points C, a, b, is calculated, and the radius, EJ, 
called " Rad.," and distance, HE, called " Dis.," are tabulated for a 
large variety of pitches and sizes of gears. 

The flank is treated in like manner for three points C, d, e with 
the radius Fd and distance Fl determined for various sizes and 
tabulated. 

Tables for unit pitch may thus be made out for all varieties of 
teeth, not only of the epicycloidal order, but involute, conjugate, etc. 

The application, in addition to drawing the pitch circle and 
pitch point, requires simply that " Dis." be taken from the table, 
multiplied by the pitch and laid off on a radius from the pitch line, 
as HE, and a circle drawn through the point concentric with the 
pitch circle. Then the "Bad.," taken from the table, multiplied 
by the pitch, is to be taken iu the dividers, when circular faces Cab 
may be struck through all the pitch points C from centers taken on 
the circle through E. 

In like manner, the circular flanks are to be struck in from cen- 
ters on the circle through F by aid of values for " Rad." and " Dis." 
taken from the table. 

The maximum error, or deviation of these circle-arc faces from 
the true epicycloid, between 6 T and b, is stated to be less than the hun- 
dredth of an inch for a pitch of three inches, a quantity hardly 
worth considering in practice, except in very large, heavy gears with 
machine-dressed teeth. 

The true epicycloidal face for Cab lies outside the circle be- 
tween C and a, and inside between a and b, and differs from the 
Willis circular face in that the latter is more nearly parallel to the 



162 PKINCIPLES OF MECHANISM. 

epicycloid from b to a and leaves a fullness at C. In action, the 
Willis teeth will thus be more inclined to receive the heavier bearing 
pressures near C and the lesser near b, or have the working contact 
between the teeth near the line of centers where the slipping action 
is least, rather than at a remote point from the line of centers with a 
greater rate of slip. 

By some it is thought advisable to arbitrarily allow the teeth to 
be somewhat slack near ab, to enable the teeth to perform with com- 
paratively light working pressures here, and heavier at or near C, 
thus reducing the strains, the friction, and the wear of the teeth. 
To this end, the radii from the Grant tables may be arbitrarily re- 
duced in length to give a corresponding slackness in the neighbor- 
hood of ab. 

In Fig. 186, the two points d and e, in the flank, chosen to lo- 
cate the flank circle arc, provide for a greater depth of flank than 
ever goes into action on a face, and it seems probable that e should 
have been chosen not much, if any, below, the working depth near 
the middle of Ce, in order to the greatest precision of the actual 
working tooth profile. As a result, the circle arc between C and d 
makes the tooth fuller than would the actual hypocycloid. 

If an " odontograph " is to mean an instrument, it appears that 
the Grant odontograph is simply any ordinary measuring scale. 
Thus the pocket rule becomes an odontograph which by analogy 
with the Willis odontograph is used in determining two distances 
on the drawing, instead of one angle and one distance. 

Circle- Arc Tooth Outlines may be determined directly from the 
rolled epicycloids or involutes; as, for instance, in Fig. 178, having 
drawn the addendum circle and supposing it to have cut near G, 
find by trial a center point that will give a circle arc that approxi- 
mates most favorably with the curve CFG. Then, noting that 
center point and radius, describe a circle through the point concen- 
tric with the pitch line, and make it the locus of the centers of all 
tooth-face circle arcs for that wheel, and, using the radius found 
above, strike in the tooth faces. Likewise proceed for the flanks. 
Involute profiles require but one center and radius. 

Co-ordinating of the Tooth Profiles has been worked out very 
completely by Prof. J. F. Klein of Lehigh University, whereby the 
correct tooth profile curve is plotted by aid of co-ordinates taken 
from a table. 

This method would seem to possess special advantages for large 
teeth where accuracy is required and where the ordinates have a 



PEACTICAL CONSIDERATIONS. 163 

length convenient to lay off. For teeth of one inch pitch or less 
a niagmfying-glass would seem to be a necessity, as well as very 
fine measuring and drawing devices. Very elaborate tables accom- 
pany the method and are embraced in Klein's Elements of Machine 
Design, 

These Various Methods of drawing the teeth have each their 
peculiar advantages, the most advisable for one case not being so 
for some others. For instance, when very large teeth are to be 
drawn, with plenty of time for accuracy, the designer may do dif- 
ferently than when the teeth are more moderate in size and more 
hurried. 

All of the above, however, admits of more or less of "hand and 
eye " operations, in which errors of greater or less magnitude will 
be unavoidable. 



CHAPTER XIV. 



MACHINE-MADE TEETH. 



Gear teeth, where no " hand and eye " process enters the ac- 
count, are possible, and in fact are in practical commercial operation 
to-day, both where the processes conform strictly with theory, and 
where they only approximate it. Doubtless watch gearing is the 
best example of the former, the hand and eye operations all being 
eliminated, as well they may be from the very fact of the minute- 
ness of the teeth. 

In Fig. 187 is illustrated what may be termed an Epicycloidal 
Machine used to cut a tool to the truly epicycloidal shape. 
Here the tool D to be formed epicycloidal is made fast to the 




Fig. 187. 

shaft G. One end of the shaft is pivoted by universal joint to the 
post F, while the other end has the disk A made fast to it. This 
disk rolls upon the stationary disk B. At E is a guiding wheel 
charged with diamond dust or emery. 

The disk A represents the pitch circle of the wheel for the teeth 
of which the tool D is to be shaped, while the disk B represents 
the pitch circle of the pinion. 

164 



MACHINE-MADE TEETH. 



1G5 



As A rolls upon B, the tool blank D is made to rub against 
the revolving grinding wheel E. By repeated rolling movements 
of A upon B, the former being shifted occasionally to bring D to 
touch the grinder E, the tool will finally become ground so as to 
touch E slightly throughout the rolling of A upon B, when the 
tool D will be finished to the true epicycloidal shape. 

The pivoting of one end of the shaft or bar G at F has the ef- 
fect of making A and B the bases of rolling cones with common 
vertex at F, so that by passing a plane transversely at the tool D 
we cut the cones in circles proportional to the circles A and B. 
Thus, the nearer the tool is to F the more minute in effect will the 
rolling circles become, so that with D at the proper point on G 
our rolling circles will be of watch-wheel dimensions, while the 
circles A and B will be large enough to manipulate conveniently. 

To prove that we thus obtain the truly epicycloidal form of 




Fig. 188 

ground curve on the end of D, we refer to Fig. 188, which repre- 
sents a section taken at D, through the tool and the cones A and 
B. Here, however, A and the tool are supposed stationary, with B 
rolling upon it, and with the grinding disk E also moving around 
with B. In Fig. 187, the grinder E is placed with its face 
coincident at the axis of the cone B, as shown in the section, Fig. 
188. Then, supposing E to move with B as the latter rolls on A. 



166 PRINCIPLES OF MECHANISM. 

and noting them in several positions as at JK, etc., we find that 
the face of E wipes up an epicycloid DJK, the same as if a 
describing circle, GL, were used with diameter equaling the radius 
of B, and with a tracing point at L. Thus the end of D has the 
epicycloidal curve DJ formed upon it, with a continuation from D 
to H on a radius of A. This line, HDJ, is perceived to be exactly 
that for a tooth profile for A, answering to the case of flanks 
radial in both pitch lines B and A. The relative as well as ab- 
solute sizes of A and B are seen from the preceding to be entirely 
arbitrary, so that profiles HDJ may be obtained for a wheel or for 
a pinion. 

The tool D thus obtained is only provisional to the final "fly 
cutter" for cutting the watch wheels, or the little milling cutter 
for the pinions; and its application is shown in Fig. 189, where D 
is used as a planing or turning tool, as the case may be, to form I, 
the final cutter for cutting the wheel A. 

The above process is much simplified by the fact of radial 
flanks for the teeth. These, for watches, 
have ample strength and are light-run- 
ning in action. 

The same process may be applied for 
wheels of any size, and when for ordin- 
/ ary machinery, cylindrical wheels, A and 
' e . B, may be used instead of the conical 
ones in Fig. 187, though for cones AF 
and BF, of considerable length and 
gradual convergence, the error due to 
the cones may be reduced to an inap- 
preciable quantity, with results which 
Fig. 189. are practically perfect. Teeth, as above, , 

are independent of all " hand and eye " 
processes such as locating an irregular curve to draw a tooth 
face, filing a sheet zinc templet to fit a tooth profile, etc. 

The teeth as produced above, all having radial flanks, though 
entirely suited to such special sets of gearing as used in watches, 
clocks, and some heavier machinery, where interchangeability is not 
sought, are not adapted for outfits of simple sets of cutters for 
general commercial purposes, either in the tool store or in the 
machine shop. For this it is generally admitted that there are but 
two systems of gearing and cutters in use, viz: 




MACHINE-MADE TEETH. 167 

The Involute and the Epicycloidal Cutters. 

These, for cutting interchangeable gearing of its own kind are 
in extended use: the first in accordance with principles explained 
in Fig. 166, and the second following the principles of Fig. 153. 

In either of these systems, cutters are made in series, each of 
which will cut several wheels of differing tooth numbers, one of 
which gears will be right, the others slightly inaccurate, but none 
out of the theoretic condition to an extent to produce appreciably 
bad-working wheels. 

The first system of cutters of this kind for cutting gear wheels 
were put out by the Brown & Sharpe Mfg. Co., and were approx- 
imately involute in form of tooth cut, each pitch embracing eight 
cutters to cut gears of from twelve teeth to a rack, and inter- 
changeable — one cutter cutting gears of twelve and thirteen teeth, 
the next fourteen and sixteen, the next seventeen to twenty, the 
next twenty-one to twenty-five, the next twenty-six to thirty-four, 
the next thirty-five to fifty-four, the next fifty-five to one hundred 
and thirty-four, the last one hundred and thirty-four to a rack. 

The second system of cutters for epicycloidal teeth of wheels in 
interchangeable series are also made by the Brown & Sharpe Mfg. 
Co., but by machinery brought out by the Pratt & AVhitney Co. 
This system embraces twenty-four cutters for each pitch, cutting* 
from twelve teeth to a rack, a wheel of fifteen teeth having radial 
flanks. For a full description of the machinery for making these 
cutters, with no "hand and eye" operation, see MacCord's Me- 
chanical Movements, page 178. 

There are two machines, one of which is called the Epicycloidal 
Milling Engine, which mills or cuts out all the tooth templets re- 
quired in the series, of magnified size; and the other is the Panta- 
graphic Cutter Milling Engine, which applies the above templets 
in making the gear-tooth cutters reduced to any desired size or 
pitch. Other templets may also be used on this last-named 
engine. 

The number of cutters in a series is arbitrary and depends upon 
the inaccuracy allowed to be admissible. For large cutters, as for 
large teeth, it would seem that the number of cutters in a series 
should be relatively great, that the admitted errors of cut teeth 
may not exceed a certain arbitrary value. A formula for deter- 
mining the so-called "equidistant series" of cutters is given by 
Geo. B. Grant in Teeth of Gear Wheels, page 20. Experience in 



168 



PRINCIPLES OF MECHANISM, 



the management of gearing would lead one to adopt a tooth 
numbering in the above series of cutters such that the cutter that 
cuts gears from 50 to 60 teeth, for instance, will make the 50- 
toothed wheel right and the others slack on the point, rather than 
the contrary, that the teeth, in action, may have the severest pressure 
of contact near the line of centers rather than remote from it, as 
explained at Fig. 186. 

Prof. Edivard Sang's Theory of the Conjugating of the Teeth 
of Gear Wheels in interchangeable series has been applied in two 
different and patented gear-cutting engines, the first by Ambrose 
Svvasey of Cleveland, Ohio, Pat. No. 327,037; and the other by Geo. 
B. Grant of Lexington, Mass., Pat. No. 405,030. 

In the first, or the Swasey Engine, a split multiple cutter is 
employed to cut the gear teeth directly; while in the second a 
solid worm or screw hob is used in cutting the gear. In each of 
these one and the same cutter cuts all gears of a given pitch, re- 
gardless of size of wheel, and giving theoretically correct teeth for 
all without the intervention of hand and eye operations. 

Let Fig. 190 represent Sang's principle in the case of Fig. 153, 
as applied to the rack where the de- 
scribing circles for face and flank 
are one and the same, carrying the 
tracer D to trace the face CD, and 
the tracer E to trace the flank CE of 
the rack tooth as the describing circle 
rolls along the straight pitch line. 
These curves for face and flank are 
both cycloids and identical in form. 

Now, as any wheel of the interchangeable series of Fig. 153, of 

the same pitch and describing 
circle, will work correctly with 
this rack, it is plain that if a 
multiple cutter were made with 
cutting teeth of the same longi- 
tudinal section as Fig. 190, that 
cutter would cut any gear of the 
series referred to, if, while cut- 
ting, the cutter could be moved 
relatively to the gear, as A is 
Fig. 191. relative to B in Figs. 123 and 

124; or as shown in Fig. 191, where D is the multiple cutter, and 




Fig. 190. 




wWS 



MACHINE-MADE TEETH. 169 

B the wheel being cut while it moves along as if its pitch line were 
rolling on the pitch line EF from E to F, the cutter at the same 
time revolving and cutting. 

This would seem to require a cutter as long as the circumference 
of the gear. To avoid this, Mr. Swasey splits the cutter, D, into 
halves, so that the idle half can move back a pitch while the other 
half is cutting and moving with the periphery of the wheel, the 
latter being kept in steady revolving motion with its axis stationary. 
In this way, the cutter, D, may be so short as to reach from the in- 
tersection of its addendum line with that of the describing circle 
on one side over to like intersection on the other side, or from G 
to H, Fig. 190. 

In cutting a gear, B, the latter is kept steadily revolving, and 
also the cutter D, in exact relation by gearing, the cutter making 
as many revolutions to one of the gear blank to be cut as there 
are to be teeth in that wheel. Then with B and D in motion in 
the cutting engine, a slow feed is given to the cutter, D, toward the 
wheel blank, B, cutting all the teeth together by one continuous 
action. When the gear is partly cut, all the teeth are cut to the 
same extent, and all are finally finished at the same time. 

To form the cutter D, Fig. 191, a tool is required which, on its 
cutting end has the exact shape of a space in the rack, or of 1C J KNL, 
Fig. 190. This tool may best be formed by some such device as 
the Pratt & Whitney Pantagraphic Cutter Milling Engine men- 
tioned above, if it is to have the cycloidal form without hand and 
eye processes. 

But in the Sangfs theory we are not confined to the cycloidal 
form of curves 10 J and KNL; for a brief consideration will show 
that any arbitrary curve 10 may be adopted and copied at CJ by 
retaining the point in common and swinging i" around to J in the 
plane of the paper, and then the whole curve 1C J turned over and 
copied at KNL. Thus, for the pantagraphic milling engine, a circle 
may be cut in a lathe for the parts 10 and CJ of the templet. 

Again, and for the simplest case under Sang's theory, 10 may 
be a straight line, which should be located at a certain angle of IJ 
with NO, usually about 14 \ degrees. Also the same for KL. 

This last-named case gives us the well-known series of inter- 
changeable wheels with involute teeth. 

This multiple cutter, thus made in two parts, is, in effect, turned 
in a lathe, by cutting one space groove after another with the tool 



170 



PRINCIPLES OF MECHANISM. 



formed as above explained, the top face of the tool being held in the 
meridian plane of D while cutting it. 

In the Grant Cutting Engine the splitting of the cutter D, thus 
complicating and weakening it, is avoided by the use of a solid 
worm cutter or hob, the latter making one revolution while the 
gear turns one pitch. In cutting a gear, the motions of the cutter 
and gear bank are continuous and in definite relation till the gear 
is finished, similarly as in the Swasey machine, and this machine 
would seem to have the preference, unless it is found that the hob 
cutter is so complex in form of worm threads as to be unduly diffi- 
cult to make. 

Investigating this, we find that hobs for cutting epicycloidal 
teeth must have the axes of the hob inclined to the plane of the 
gear being cut by an angle equal that between a plane normal 
to the axis of the hob and the tangent to the worm thread of hob, 
at the pitch line. That is, the element of thread of hob, at the 
pitch line, must be perpendicular to the plane of the wheel. This 
inclination can easily be found from a triangle where one side is the 
circumference of the hob at the pitch line, another side, perpendic- 
ular to the first, the pitch; in which triangle the smaller angle is to 
be taken for the inclination. 




\ 6 


\r, J 


v K-,d 




Ulli 



G 



Fig. 192. 

Even with this inclination of the thread of the hob (speaking of 
it as before the teeth are cut in it and when it is simply a screw), it 
will not be tangent to the cut teeth of a gear, or of a rack, Fig. 190. 
cut by it, except in an S-shaped curve, acl, Fig. 192, which is more 
pronounced as the radius r is shorter and not vanishing until r Is 
infinite.* 



* The equation of this curve acb is r sin 



- AB + Vl + -S 2 - A 
l + #* 



in 



MACHINE-MADE TEETH. 171 

To give an idea of the intensity of the S-curve for epicycloidal 
teeth, take r, = 2",p = 0.5". Then db, Fig. 192, is 0.158"; and 
for the ten equal divisions from d to a, the remaining ordinates are 
0.149", 0.126", 0.104", 0.075, .0", - 0.078", - 0.107", - 0.129", 
— 0.147", and — 0.162", the extreme ordinates being the greater. 

The tool which cuts the worm, as in a screw-cutting lathe, 
should have this shape on its top side at the cutting end, as shown 
dotted at G and H> in case the tool is to be used as in cutting other 
threads. But a preferable way is to make its top flat and straight 
and of the true epicycloidal shape in plan as at H, and, after cutting 
in the ordinary way to the right depth for the hob thread, then, 
without varying the depth of cut, make several passes for cuts with 
the tool at varied heights, raising and lowering on a line which is 
tangent to the hob thread at the pitch line, and for a range cover- 
ing the ordinates db and ea. A safe way would be to work the tool 
at different heights thus as long as it will take a cut. This sup- 
poses that the tool for cutting the hob thread has the correct shape 
as at ICJKNL, Fig. 190, and formed as already explained. 

But in cases where the tooth profiles are not normal to the 
line CN, Fig. 190, the cutter hob will not need inclining so much, 
and, in some cases, not at all, as for the involute series. 

The hob for involute toothed ivheels in interchangeable series for 
use on the Grant Cutting Engine is to be made in a similar way 
if its axis is to be inclined as before, so that at the finishing of cut 
the tangent to the pitch-line element of the hob thread is normal to 
the plane of the wheel being cut. But for this case of involute 
gearing it is not necessary to incline the axis, as it may be kept 
parallel to the face of the wheel being cut, if the hob is formed 
with due regard to this, viz. : that, in cutting the hob threads, as in 
a threading lathe, the tool be held at varying heights as before, 
while cutting, except that now it is to be raised and lowered on a 
vertical line instead of a tangent to the pitch-line element, and to 

which, for epicycloidal series, A — — and B = — J A/ r ~" Tl i n which 

r p V 7.5p — r-f ?'j' 

last, p is the diametral pitch. 

In involute series with axis of hob inclined as above stated, A = — , and 

r ' 

B = — • tan 141^ = 0.52 — . In constructing these curves, r is to be laid 

off from D on Be ; and r sin'Q is to be laid off from Be on circle arcs struck 
from B. 



172 PRINCIPLES OF MECHANISM. 

•continue the raising or lowering and repeated cuts as long as it will 
take cuts. In this case, however, the tool may be placed at the 
height, first above and afterwards below the axes of the hob, in 
cutting the thread equal the diametral pitch divided by twice the 
tangent of the angle of inclination of the edges of the tool, or of 
the sides of a tooth in the rack, which inclination is usually about 
14|- degrees. Hence the height above or below is about twice 
the diametral pitch. 

In all cases above, the thread tool for cutting the hob threads is 
to be correct in shape and dimensions as answering to the outline 
ICJKNL, Fig. 190, except that it should be somewhat in excess of 
length so as to cut the hob deep enough to have clearance at the 
top of the teeth, and with due regard to side clearance between re- 
sulting teeth of wheel. 

It appears that the hob threading tool must have the correct 
shape in any event according to the system, and of the right thick- 
ness for a tooth; but that the threads of the hob are not of correct 
tooth profile shape in any case. For involute teeth these threads 
will not be straight on the meridian intersections. Hence teeth 
of spur wheels cut in this way in the Grant Cutting Engine with 
the ordinary worm-wheel hobs will not be correct in profile of 
tooth. 

In these gear-cutting engines of the Swasey or Grant order 
the spacing of the teeth may be expected to be unusually even and 
exact, since in the cutting the hob acts upon several teeth at the 
same time, and the wheel being cut may be driven around by worm 
and wheel. The latter may be made in two half wheels joined on 
a plane transverse to the axes and through the middle of the teeth. 
In cutting this duplex worm wheel, when partly cut it may have 
one half loosened and shifted on the other half the space of a few 
teeth, made fast and some further cutting done, then shifted again 
and again, till the teeth in the two halves will all fit exactly for 
any way of combining the halves. Then when in use and wear the 
part may be occasionally shifted. In this way, with these cutting 
machines, it seems certain that the spacing as well as the tooth 
outline of cut gears may be made marvels of exactness. 

Tooth Planing or Dressing Machines. 

These machines, like the Gleason's, are in use in some machine- 
shops, where the teeth of large heavy cast gears are tool dressed, as 



MACHIKE-MADE teeth. 173 

in a shaping machine, the tool being guided by a templet which 
may have been formed by hand, thus admitting hand work in part. 

George H. Corliss appears to have been the first to do this, his 
first work dating back to the 40's or 50's: the heavy fly-wheel 
gear on the Centennial Corliss Engine, central in Machinery Hall, 
being one of the more notable examples of tool-dressed spur wheels. 
Large bevel wheels, connecting the main shafting with the above 
engine were also tool dressed on a bevel " gear planing " machine 
exhibited by Mr. Corliss. 

Hugo Bilgram exhibited remarkably smooth-running bevel- 
gears at the World's Fair of 1893, the teeth of which were dressed 
out in a similar way, as an example of commercial work by him. 

In these tooth-planing machines for bevel gears the point of 
the cutting tool is made to move on a slide in a line joining the 
vertex of the cone of the gear and the point of contact of the guide 
finger with the guiding-tooth templet, this tool and finger being 
mounted, in effect, on a universally swinging arm, pivoted at the 
cone vertex. Thus the elements of the finished tooth are made to 
converge to the apex of the conic gear. Large or small gears can 
be tooth dressed on the same machine. One considerable advan- 
tage here over the spur-gear tooth-dressing machine is that the 
tooth templet may be made of magnified size, so that bevel wheels, 
compared with spur wheels thus tooth-dressed, may be regarded as 
more nearly perfect, while the contrary is true of wheels cut in 
the ordinary way with revolving cutters. 

Stepped and Spiral Spur Gearings. 

In some of the finer light-running machinery at high speed the 
plain spur gearing is quite likely to make a humming sound as the 
teeth engage in succession, each contact giving a slight click. 

This is avoided in a measure by stepped gears, the wheel being- 
made up of several thin ones made fast together and arranged in 
steps so as to divide the pitch into as many parts as there are thin 
parts in the wheel. 

Another way is to use the spiral gearing of Hooke, in which 
each tooth is on a spiral slant, to such extent that one pair of teeth 
engage before the preceding pair quits engagement. 

This causes endlong pressure on the axes, objectionable in 
heavy working wheels, but which is often obviated by making each 
tooth of equal portions of right and left handed spirals. 



CHAPTEE XV. 
SECOND: AXES MEETING. 

BETEL GEARING: 

The teeth of these wheels may be made of any of the forms ex- 
plained under the case of axes parallel. The pitch surfaces being- 
cones, with vertices at a common point, the describing curves, by 
analogy with cylindric gears, are also cones, with vertices in com- 
mon with the pitch cones. 



Theoretically Correct Solution. 

In practice the bases of the cones may be made spherical, with 
centers at the cone vertices, and the describing cones ma}' be 
realized in concave templets fitting these spherical surfaces, as 
explained in Fig. 138. In this way, the correct tooth curves may 
be laid out on the gear blanks, the number and kinds of templets 
required being explained in Figs. Ill and 112, though here all 
made concave and fitting the spherical base of the cone. 

In these wheels, no circular rolling cone can describe precisely 
radial flanks, and to approximate them the describing circular 
cone must be made a trifle larger than to span the distance on the 
sphere from the pitch line to the pole or axis to the wheel. 

Approximate Solution. 

But a much simpler way, and that usually followed in practice, 
is Tredgold's approximate construction, which, though slightly 
inaccurate in theory, is appreciably exact in practice. In Fig. 193, 

174 



AXES MEETING. 



175 




Fig. 193. 



take A and B for the wheels with 
pitch surfaces in contact at CG, they 
being of conical form, with the cone 
vertices at a common point in 0, the 
axes being OAD and OBE. The 
length CG is arbitrary, and also the 
angle of intersection of the axes at 0. 
In practice the latter is usually a 
right angle. 

Tredgold draws a line, DCE, per- 
pendicular to the line or element of 
contact CO of the pitch cones, when, 
if CD be revolved about the axis AG, 

a conic surface A CD J will be generated, which is normal to the 
pitch cone, ACOJ. Likewise will the line CE generate the cone 
BCEK, normal to the pitch cone BCOK. Developing these cones 
from the line ED, we obtain the circles of development DCF and 
ECH, with centers at D and E. 

Upon these circles the teeth are laid out as if they were the 
pitch lines of a pair of gears, any system of teeth being selected 
as preferred. For epicycloidal teeth with concave flanks, the 
describing circle carrying the tracer J has a diameter which is less 
than CE, and in the usual way is rolled inside of CH and outside 
of CF, to generate a pair of mating tooth curves, a face and flank. 
Another describing circle is rolled on the other sides of the circles 
CF and CH, thus completing the tooth profiles as dotted. 

The addendum circles, root circles, clearing curves, etc., may 
be drawn in, as in spur gearing, when the conic surfaces CD F and 
CEH may be re-developed or returned to the cones CD A J and 
CEBK, taking the tooth drawings with them, giving us the laid-out 

teeth on the conic blanks, as 
in Fig. 194. 

In preparing the blanks, 
they should be left larger to 
include the addenda and other 
portions of the finished wheel 
as shown in Fig. 194, unless it 
is preferred to dress the flank 
to the root surface and add 
the teeth thereto as often done. 
;ill vanish at 0, and a convenient 




Fig. 194. 
The tooth-surface elements should 



176 PRINCIPLES OF MECHANISM. 

as well as sure way to give the teeth the right directions in the 
finished wooden gear patterns, for instance, is to fix a fine line at 
the point 0, which may be drawn to the tooth outline at C to 
determine when the tooth is dressed to the proper lines, element 
by element. 

Bevel wheels are not practicable in interchangeable series, 
because, if one pair have axes at right angles, the substitution for 
one of these of another correct working wheel of larger or smaller 
radius changes the angle between the axes to something other than 
the usual 90 degrees. Therefore the teeth, and both wheels entire, 
must be made in pairs and of shapes to suit, regardless of inter- 
changeability. 

Spiral Bevel-wheel Teeth. 

These are possible, but not common, because difficult to make, 
except by special machinery, which probably does not exist. Stepped 
teeth would be more readily made. 

Bevel -wheel teeth, carefully planed to shape by templet in a gear- 
planing machine, will work fairly well. See Tooth Planing, under 
Axes Parallel. 



CHAPTER XVI. 
THIRD: AXES CROSSING WITHOUT MEETING. 

Skew-bevel Gear Wheels. Approximate Construction. 

As stated in connection with non-circular skew bevels, a method 
of laying out theoretically correct and practicable teeth is not 
known, except for gears taken at or near the common perpendicular 
between the axes, or near the gorge circles. 

But teeth which approximate the epicycloidal form may be 
generated by employing generating hyperboloids to roll upon the 
pitch hyperboloids inside and outside, in a manner analogous to the 
use of describing cones in common bevel wheels, and as shown in 
Fig. 138. These approximate tooth curves and surfaces become 
more and more inaccurate as the angle between the axes increases 
toward 90 degrees, at which limit considerable interference occurs, 
requiring the teeth to be " doctored " by arbitrarily dressing off 
certain tooth faces or mating flanks to an appreciable extent, to get 
the wheels to work with acceptable smoothness. 

The patterns for cast gears may thus be executed to better ad- 
vantage, probably, than in any other way, even not excepting Prof. 
MacCord's construction, based partly on Olivier's theory of invo- 
lutes for one tooth surface, assuming another, and determining the 
rest by difficult conjugating. 

To construct these approximate skeAV bevels, let PRLNSQ, in 
Fig. 195, represent the blank of a skew-bevel wheel, for which the 
hyperboloid of revolution 1TUVKM is the pitch surface, deter- 
mined as in Fig. 10, extended from IM to and past the gorge circle,, 
TK, reaching £7Fat a distance OZ ' = OA beyond the gorge circle. 
The addendum and dedendum surfaces PQ and RW are drawn 
in throughout, being hyperboloids of revolution, because a top cen- 
ter line of an extended tooth, as well as a root line, being straight 
would, in revolving, sweep up hyperboloids of revolution, for the 
same reason as would the element CO of the pitch surface. At 
JK these three lines are shown as parallel to the axis in projection, 
which determines the height, ab, of a tooth at the gorge circle. 

177. 



178 



PRINCIPLES OF MECHANISM. 



Then the contour lines of the addendum surface can be drawn in 
as hyperbolas, NSbd and LYae. The lines NL and SY should be 
extended normal to MK, to the bottom of the web upon which the 
teeth are mounted. With this much drawn, the skew-bevel blank 
can be turned up, arms and hub being assigned at will. 




Fig. 195. 

Tredgold's method is adopted, as in Fig. 193, of developing in 
IF the normal cone (normal to the pitch surface) upon which to 
lay out the teeth at F. The epicycloidal or involute form of tooth 
may be chosen here, as well as in ordinary bevel gearing, but this 
example will be carried out in the approximate epicycloidal form, 
with a view to testing the theory of the generating hyperboloids. 

The rolling circle applied here gives a close approximation to 
the normal section of a tooth at 7, as would be developed by the 
describing hyperboloid above explained; so the full drawing of the 
skew tooth will be first given, and its errors afterwards sought out. 



AXES CROSSING WITHOUT MEETING. 179 

Instead of the normal section of tooth, we require an oblique 
section, as shown at A, formed by the intersection with the tooth 
of the surface of revolution NLRP. To obtain the excess of 
width of this oblique over the normal section, revolve the line CO 
(same as CO, Fig. 10) about the axis, till C falls at A and at E. 
Then revolve the center line EA of the tooth about E, till A falls 
at G, where EG is parallel to the plane of the paper, when EG 
appears in its true length. A convenient way to find G is to make 
EG equal in length to CO, G being on the line AD. Then the 
angle OGE is the angle of obliquity of the normal section of tooth 
with the oblique section at A, Therefore the width of the oblique 
compared with the normal section of tooth at i^is in the ratio of 
GE to GO. This may be laid off at several points in the height 
and the oblique section determined. If preferred to do it now, the 
faces and flanks may be arbitrarily doctored from the pitch line, 
each way, to prevent interference, as explained above in connection 
with Fig. 196. 

With the oblique section determined, as in the outside lines 
at F, we may cut a templet in thin material and use it on ICM, as 
explained in Fig. 179, to delineate all the teeth on the pitch line 
IM. 

Then, in cutting the teeth to shape, the oblique direction AE 
is probably best arrived at by carefully cutting a thin piece UV to 
all the tooth outlines, beveled back in excess from the tooth profiles, 
and mounting it on the axis AOZ so that OZ equals OA, and in 
such position that the line CO from the center of a tooth at C 
will strike the center of an outline at H. Then a thread may be 
drawn at any time, in dressing a tooth, from any point on H to the 
like point on C, and each element of C thus dressed the whole 
length of the tooth to fit the line. This process, repeated for all 
the teeth, completes them unless when the mating wheel is likewise 
thus far completed and the pair be placed in running relation it 
be found that interference exists. If so, the teeth are to be dressed 
off at will, till interference is relieved, which completes the wheel. 

These wheels may thus be made of any desired length from C 
toward H. To give an idea of the amount of interference of the 
epicycloidal teeth as above, Fig. 196 is introduced, which was drawn 
with great care and labor from a model like that described by Geo. 
B. Grant in the American Machinist for Sept. 5, 1889, Fig. 3. In 
the present example the axes are at right angles, at a distance of 3.7 
inches apart at the common perpendicular, the wheels being 11 and 



180 



PRINCIPLES OF MECHANISM. 



6.8 inches in diameter respectively, and with the velocity-ratio of 
3 to 2. These tooth-curves were traced with the true rolling hyper- 
boloids. 

The figure is a transverse section, taken on a perpendicular to 
the element of contact, CO, Fig. 195, and at a distance of 5.8 inches 







=— - —Ik \ \ 


^^/^\ 


^^ 






=^l\\ /\\ ,\\ 












A^ 


^JU^^^X^ 


^y 



Fig. 196. 

from 0. It cuts through the two pitch hyperboloids in curves of 
intersection A A and BB, and the describing hyperboloids in the 
curves DCE and FCG, which curves of course are all ellipses. 
Also it cuts all the mating tooth- curves at three quarters of an inch 
apart on the pitch line. 

The describing hyberboloids are both of one size, and half as large 
as the smaller wheel, so that by theory the flanks in the smaller 
wheel should be radial. The figure shows them to be very nearly so. 

This is a somewhat extreme case of proportions, and we might 
expect high per cents of interference. The figure shows almost no 
interference on the left-hand side of the pitch lines up to a pitch 
of over two inches; while on the right-hand side, for a pitch of f 
inch, H inch, and 2\ inch, we find the faces and flanks cutting into 
each othei to a normal depth of 5, 6, and 10 hundredths of an inch 



AXES CROSSING WITHOUT MEETING. 181 

respectively, or about four per cent, of the pitch. For a pitch of 
eight inches the interference depth reaches 15 hundredths of an 
inch, or about two per cent, of the pitch. 

Also, the figure shows that below C the faces of the smaller 
wheel must be trimmed off toward the point and on the lower side 
ior pitches over about 1.5 inches, while the flanks of teeth of the 
larger wheel above C seem to need dressing out on the lower sides. 
The figure may thus aid in determining how to doctor the teeth in 
a practical case, for best. results. 

One noticeable feature of these tooth curves is that, if they 
interfere, they intersect at the surfaces of the describing hyperbo- 
loids, in the central positions shown, viz. : on the lines ECD and 
FCG, — a fact also pointed out by Grant in 1889 in the few curves 
shown. 

Combined punching and shearing tools have had skew gears 
some 20" diameter. 

Exact Construction. 

The exact construction of these teeth has been best treated by 
Grant in the American Machinist and also in his valuable work on 
Teeth of Gears, where the theory of Olivier and treatment by 
Herrmann are discussed. 

Olivier appears to have originated involute tooth surfaces, which 
he calls spiraloids, that may be generated by the revolution of a 
straight line about a cylinder, the line being maintained at a con- 
stant angle with the cylinder and prevented from lateral slip. The 
same result is accomplished by rolling a plane around a cylinder 
without slipping, on which plane a line is drawn obliquely. This 
line will sweep up, in the space about the cylinder, the spiraloid of 
Olivier, which spiraloid surfaces are proposed for skew-bevel teeth. 

Any point in the above line will evidently describe an involute 
about the cylinder, and all points of the line 
together, will describe an involute spiral or spir- E 
aloid. 

In this description it is immaterial to the 

result if the plane slips on the cylinder in the 

direction of the line, 'that is, if the line be only 

allowed to slide on itself, not laterally, and the 

angle between the line and cvlinder be preserved 

& , , " Fig. 197. 

constant. 

Fig. 197 shows a section at right angles to the spiraloid as 




182 



PRINCIPLES OF MECHANISM. 



swept up by one line on the plane, CED being cut by the line at 
one side of that point of the line which comes to touch the cylinder 
A at C, and CFD cut by the line at the other side of the same point* 

The Olivier Spiraloid. 

In a general view this spiraloid with section, Fig. 197, appears 
like a twisted bar in which the cylinder A C is straight, the de- 
pression C being like a spiral or helical 
'crease along the length, and the edge D like 
^j a spiral ridge. According to what is stated 
above, the curves of cross-section CED and 
CFD, Fig. 197, are the ordinary involutes to 
the circle AC. 

In Fig. 198 two views are presented of a 
multiple spiraloid of six ridges and creases, 
'the intersection of each ridge by a- normal- 
plane giving two equal limited involutes, as 
CD and ED. Extended involutes give Fig. 
203. 

A straight line, FGH, will touch one 
ridge from F to G, while from G to H it lies 
in contact with the under side of the next 
ridge above FG. Therefore these ridges 
are enveloped by surfaces which have in- 
volute elements in normal section and right- 
lined elements in lines tangent to the cylinder 
Fig 198 ACEJ, as in the example of. the line FGH. 

Interchangeability of the Olivier Spiraloids. 

Now if we remove the alternate ridges or threads, and cu^ 
clearance grooves into the cylinder to sufficient depth, as shown 
by the dotted lines UK, this multiple spiraloid will work, accord- 
ing to Olivier, as a skew-bevel gear with another like it, or with 
any other of the same normal pitch, regardless of the axial or the 
circumferential pitch of the helical threads or teeth. This is true 
even at the limit where the helical ridges become parallel to the 
axis, provided that the normal pitch is still the same and that the 
normal planes cut the ridges in involute lines of section, as at CD y 
Fig. 198. 

This is equivalent to saying that a numerous set of these spira- 




AXES CROSSING WITHOUT MEETING. 183 

loids of the same normal pitch will work together interchangeably. 
Herrmann pronounces against this. 

Interference of Olivier Teeth. 

But Olivier's claims can be shown to be true for these gears as 
cut off at the intersection of the axes, as shown in Fig. 199, when 
so cut that no contacts of teeth occur outside of the angle A OB, 
because interference of teeth takes place outside these limits, of the 
same kind as found to occur between involute teeth of cylindric 





Fig. 199. Fig. 200. 

gears with involutes drawn from the pitch circles instead of base 
circles within. An example is shown in Fig. 200 at D. 

By undercutting the teeth in the vicinity of 0, Fig. 199, and 
to some extent up the face CD, Fig. 198, with plenty of bottom 
clearance, the teeth will work when the gears extend both ways 
past 0, Fig. 199, if the addendum is not excessive, as has been 
shown by Oscar Beal in good working gears of the kind in metal. 

Nature of Contact of Olivier's Teeth. 

Herrmann claims that these gears, if working at all, will have 
tooth contacts only at points instead of lines, but it can be shown 
that the contacts will be in lines like FG, Fig. 198. 

Olivier holds that the gears as cut off at 0,Fig. 1 99, will drive only 
in one direction, and that to drive both ways they must be extended 
"beyond 0. This is proved by the Beal gears to be true, as well as 
the necessity for clearance for thick addenda, as above explained. 

Thus we have very satisfactory and perfect working skew-bevel 
gears of the kind shown in Fig. 198, with straight line contacts be- 
tween good shaped teeth, though the gears extended both ways have 
not contacts far from the gorge circles, but are capable of driving- 
either way. These gears appear, however, like cylindric skew gears, 
but are in reality skew bevel, as taken at the gorge of the hyperbo- 




184 PRINCIPLES OF MECHANISM. 

loids, though with no fixed relation between the gorge radii, and 
.angles between axes and Hue of contact. 

If the gears are at right angles the working straight line contacts 
will all be within the rectangle abcO for driving- 
one way, and in the rectangle defO for driving 
the opposite way, while the appreciable inter- 
ference will be outside the dotted curves g and 
h, as shown near 0. The rectangles are de- 
termined by the intersections of the adden- 
dum surfaces with the common tangent plane 
between the two cylinders A and B, which 
cylinders form the bases of the involute teeth. The greatest work- 
ing length of gear A is therefore ce, and of B it is bf. 

Results from an Example. 

To better fix the ideas, a particular example is referred to of the 
above cylindric skew-bevel screw-gears, in which the gorge-circle 
cylinder pitch surfaces extended indefinitely each way, as shown in 
Fig. 202, are 5.1" and 2.2" in diameter, with axes at right angles. 
For an addendum of 0.32" the interference was found inconsidera- 
ble, but for thicker addenda appreciable interference would begin 
at about the dotted lines g and h, Fig. 201, when gO and hO equal 
about 0.8". Then, in case of the thicker addenda, with clearance for 
interference cut from C up toward D, and E up toward D, Fig. 198, 
on the face the necessary amount, it is found that there will be no 
contacts at or about 0, Fig. 201, for a distance from O of about the 
same value as the distance from the dotted curves g and h over to 
where the interference for the actual addenda ceases, and probably 
within some such shaped outline as the curves g and h. These 
quantities are admitted to be only approximate. 

We may Demonstrate the Principle respecting the above exact 
construction of Olivier by use of Fig. 202 — in some respects the 
same as one given by Grant. Here, for the pitch surfaces of the 
proposed skew gears, we take AD and BE, the cylinders of the 
gorge circles, touching at 0, between which cylinders in common 
tangency is a plane GG, upon which are parallel lines, the normal 
pitch of teeth apart. Now suppose the plane to be moved along 
with its parallel lines maintained parallel to a fixed line, and that 
the cylinders AD and BE are revolved by the plane in such a way 
as to allow no lateral slipping of the parallel lines of G on the 



AXES CROSSING WITHOUT MEETING. 



185 



cylinders, while at the same time admitting endlong sliding of the 
lines in their own tracks on the cylinders to any extent. It is plain 
that if the lines could leave an impression of themselves on the 
cylinders the latter would appear like perfect screws, having a num- 
ber of linear threads. 

Now suppose that the cylinders had been covered with some 
easily cut material during this rolling, and that the material within 
the surfaces cut by the lines on G remained attached to the cylin- 
ders. It is plain that the cylinders with these thread ridges thus 




Fig. 202. 



formed and attached would be like that of Fig. 198, the surfaces 
of the threads being of involute form in normal actions and with 
straight-lined elements in all lines tangent to the cylinders, as for 
FGH, Fig. 198. Also, it is easily seen that these surfaces are tan- 
gent to each other within the angles AOB and DOE above and be- 
low 0, and that these lines of tangency will all be in the plane G, 
because this plane is normal to all the cut surfaces. Theoretically, 
then, these threads, or spiral ridges, will have perfect right-lined 



186 PRINCIPLES OF MECHANISM. 

contacts within the angles AOB and DOE to any extent from 0, 
and exactly the form of contacts required for gear teeth. 

In the above, no particular angle has been assumed between the 
cylinders, nor between either one of them and the cutting lines on 
the plane G. It appears, then, that the cylinders may have any 
possible angular relation with each other, or with the lines on the 
plane G; and that if the lines be assumed as running from into 
the angle AOE instead of the angle AOB the contacts will all be 
within the angles AOE and BOD, so that the case is perfectly 
general. 

In all cases there will be some degree of interference, as 
explained at Figs. 199 and 201, in the angles opposite those within 
which the contacts are found. 

On account of this interference it is advisable in practical 
gearing of this kind extending past that the addenda be com- 
paratively light — say in the neighborhood of a fifth of the mean 
gorge radius — and the teeth correspondingly small. 

In these gears, though perhaps called skew bevels, if the teeth 
extend to sharp tops throughout they will be cylindrical screws, as. 
in Fig. 198, there being alternating ridges or teeth, and spaces, as 
at D and 1. 

In practical gearing, the angles between the lines of G and axes 
A and B must be such that the normal pitch will divide the pitch 
cylinders without remainders. 

Gears of this kind, instead of being taken at the gorge circles 
0, may be selected at a distance from the gorge circles, as at A' and 
B' , Fig. 202, and comparatively short. But if the teeth go to sharp 
tops they will be short, many-threaded, cylindric screws. The 
teeth of these gears will be comparatively flat and practically 
useless if selected far from the gorge, though they will work per- 

I — 1-| 1 1-| — | fectly in theory. The gear A', Fig. 202, is 

shown in Fig. 203 with skew omitted, where 
A J is the cylinder AD of Fig. 202, and A' 
the gear. 

The involutes, as extended, intersect at 

^ A-i A Ly^) various points M, N, etc.; and the larger 

x ^r^J_j£ gears, like A', require no bottoming clear- 

\ ^^/^ / ance, such as needs to be provided for gorge- 

L circle gears, as at UK, Fig. 198. 

Fig. 203. j£ fi nr ,Hy appears that we have no exact 

theory for skew-bevel teeth that are of practicable utility, except 



AXES CROSSING WITHOUT MEETING. 



187 



at or near the gorge circles, those selected at a considerable dis- 
tance from the gorge being worthless from excessive flatness of 
teeth, and doing more crowding than driving. Good practical skew 
bevels must, therefore, come from the approximate solution of 
Fig. 195, or be fitted with the assumed and conjugated teeth of 
MacCord. 

It is plain that the velocity-ratio changes with a change of the 
inclination of the lines on the plane G, Fig. 202. This controverts 
the statement in Prof. MacCord's Kinematics, page 365, respecting 
twisted skew wheels. 

The Olivier Skew Bevels find application in such examples as 
Figs. 204 and 205. To design those of Fig. 204, draw the axes AO 





a 


V 


to 




w 


V 




o\ 




b ^\p 


ZLk/ / 


<§Ns 


7>v/t> 




^A 




Fig. 204. 



Fig. 205. 



and BO, and generating line DO, as answering to one of the lines on 
the plane G, Fig. 202. Then on a line perpendicular to BO lay off 
the distances OO and Oc, equal to the number of teeth A T and n in 
the large and small gears respectively, multiplied by the normal 
pitch. Draw the triangles OCa and Ocb, right angled at C and c, 
and with Oa and Oh perpendicular to the axes. Then the diameters 
or radii of the wheels will be in the relation of the lengths Oa and 
Ob. 

In Fig. 205 one wheel is a spur gear, answering to the case in 
the Olivier gears where the lines on plane G, Fig. 202, are parallel 
to one of the axes. The diagram is lettered for similar quantities 
as in Fig. 204. 

To cut these teeth, so as to make the wheels Olivier wheels, the 
cutters must be of such special forms as to give to the teeth, in 
sections normal to the axis, the involute form outside the pitch 
lines. But this is hardly to be expected in practice, and ordinary 



188 PRINCIPLES OF MECHANISM. 

cutters will be used, even if the teeth have contacts at points 
only. 

In Figs. 204 and 205 either or both wheels may be enlarged to 
the rack, which in action will slide endlong in their own tracks. 

The Worm and Gear may be made as Olivier gears by making 
one wheel very much smaller than the other, and giving the smaller 
one only one ridge or screw thread, by properly inclining the lines 
on plane 67, Fig. 202, and by use of the proper shaped cutter. 

But this, again, can hardly be expected in practice, as good 
working worm wheels are produced by using a hob cutter of the 
same size and shape as the worm itself to finish cutting the gear. 

The normal pitch should be the same here at the pitch lines 
for the worm as for the gear. When the axes are at right angles, as 
usually the case, the pitch of the worm in a direction parallel to the 
axis is the same as the circumferential pitch of wheel at pitch line. 
The wheel teeth, instead of being cut straight, are concave, as 
formed by the revolving hob cutter, thus securing more bearing 
surface between teeth. 

In a section taken by a plane normal to the wheel axis and con- 
taining the axis of worm, the shapes of the teeth of worm and wheel 
should be the same as in the rack and pinion of Figs. 162 or 167, 
with the same points observed as to interference of teeth. The 
worm may be single or many threaded. 

The Hindley or " Corset-shaped" worm may be supposed to 
possess advantages over the cylindric worm when examined in sec- 
tion; but a little consideration will show that its tooth contacts with 
the mating wheel are more like points, or, at best, a line on each 
tooth from top to bottom, because of the continually varying diam- 
eter of the worm, so that no thread can fit the teeth of the wheel 
as well as the cylindric worm thread can, with line contacts along 
the thread from side to side of wheel; and second, that the larger 
average diameter of the worm will give rise to more rapid wear. 

Skew-bevel Pin Gearing is possible for all angles between axes, 
but the pins take complex shape, except for the one case of right- 
angled axes and velocity-ratio unity; when the pins are cylindrical 
and in diameter equal the shortest distance between axes. 

Worm pin gearing is also possible; but as these pm wheels are 
scarcely demanded in practice, they will not be described here. 
Detailed descriptions are found in MacCord's Kinematics, pp. 284- 
293. 



axes crossing without meeting. 189- 

Intermittent Motions. 

Movements of this class are given in Figs. 14 to 16, where the 
pitch lines were discussed. The teeth on these wheels are the same 
as those already considered, for both the principal circular arcs and 
the non-circular initial arcs, and also for the starting and stopping 
accessories, as explained in Figs. 140 to 143. 

These movements may be made for the three cases of (1) Axes 
Parallel, (2) Axes Meeting, and (3) Axes Crossing Without Meeting ; 
but, as the principles for all these have been duly considered for 
non-circular pitch lines, they will not be taken up here for the? 
simpler case. 



CHAPTER XVII. 



ALTERNATE MOTIONS. 



L Limited Alternate Motions. 

DIRECTIONAL RELATION CHANGING. 

First. With Solid Engaging and Disengaging Parts. 

In Fig. 206 is illustrated a movement of this class with engaging 
and disengaging features, all designed with a view of being a 
thoroughly practical and durable working movement. It is laid 
out with involute teeth. 

If revolving right-handed, before the tooth c can strike upon e, 
the tooth a will have moved to some extent upon its mate b and to 




Fig. 206. 

have thrown the rack fully, so that interference cannot occur at ce. 
Then, to start B gradually, the portion at d is carried up near to 
the center of A, so that the first impulse for moving B is received 
at d, where the velocity of A is less than at a, and thus reducing 
the initial blow. This may be carried still farther toward the axis 
A if desired. 

In this construction there will be a momentary pause of the 
rack at the end of the movement. 

190 



ALTERNATE MOTIONS. 



191 



Second. With Attached Engaging and Disengaging Spurs. 

In Fig. 207 the movement has attached spurs to control the re- 
versal of movement so that the blow may be reduced. This is taken 
from a working design. The teeth are epicycloidal, though they 
may be of any form preferred. 




Fig. 207. 

The shock due to initial motion is reduced to a minimum by 
carrying the spurs up so far that the normal to the curves of spur 
and pin will strike so close to the axis of A as to indicate easy 
starting. 

For simplicity, the movement of Fig. 206 has the advantage, 
though this will be accompanied with less shock at reversal. The 
spurs may be carried still higher to farther reduce the shock. 

The Mangle Wheel. 

Under velocity-ratio constant the pitch lines must be circular, 
and in this movement the wheel is limited to one revolution at 
most, an example of which is shown in Fig. 104. 

In this form of axes meeting the wheel may have ordinary teeth 
on the edges of a band instead of pins ; but for a face wheel pins 
would be necessary, otherwise the opposite sides of a band bearing 
teeth would give different radii of pitch lines, and hence different 
velocities for forward and return motion. This construction may, 
however, be required, giving a case of velocity-ratio changing, 
though constant for each direction. 

II. Unlimited Alternate Motions. 

These are found in the mangle rack, which may be conceived 
of by supposing the wheel of Fig. 104 to be cut at the center of the 
H-shaped reversing piece, then straightened out to a rack, and 
then mounted on proper slides. 

The teeth may be on the upper and lower sides of a rack bar 
instead of a row of pins. Any length may be given to the move- 
ment, and hence it may be classed as unlimited. 



CHAPTER XVIII. 
CAM MOVEMENTS. 

In a cam movement the path and the law of motion of the fol- 
lower may be determined independently of the driver, when the 
latter is made to conform therewith. 

Thus the follower path may be a straight line, a circle or any 
other curve, and the movement of the follower in that path may be 
assumed point by point for the full forward and also for the back- 
ward movement. 

The cam is accompanied by an undue amount of friction by the 
rubbing of the follower on the driver, which in turn contributes to 
the wear of the parts, causing backlash and, in high speeds, noise. 

This movement is the one usually called to his aid as the last 
resort, when the designer, seeking to avoid it, fails to obtain the 
necessary motion of a piece by other means, such as may be pro- 
posed for lighter and more quiet running of parts. 

The cam is, therefore, a most useful movement, and in certain, 
cases must be accepted, though to be avoided whenever possible. 

CAMS IN GENERAL. 

By Co-ordinates. 

The driver may have a non-uniform motion as well as the fol- 
lower. To unite all questions in a generaL 
solution of a cam and follower in one 
plane, let A, Fig. 208, represent the driver 
axis, to which is attached a pointer, Al, 
the same turning so as to be in positions 
2, 3, 4, etc., at the ends of equal successive 
intervals of time. The unequal angles at 
A indicate variable motion of A. Let the- 
^- curve 1, 2, 3, etc., at D represent the path 

Fig. 208. of the follower, D the point of the latter, : 

which reacts against the cam, being at positions 2, 3, 4, etc., at the 
ends of equal successive intervals of time, these intervals being the 

192 





CAM MOVEMENTS. 193 

same as for A. The mechanism by which D is mounted, compel- 
ling it to move in the path 1, 2, 3, etc., is not shown. 

Draw a circle arc la from the follower path to the position 1 of 
the driver pointer. Likewise arcs 2b, 
3c, 4:d, etc., as shown. This gives us 
co-ordinates by which to construct the 
cam, as in Fig. 209, where a\, b2, c3, ^ 
etc., are laid off from the initial line 
Al. 

Then the cam outline may be drawn Fig. 209. 

in as the curve 1, 2, 3, 4, etc. In practice this may all be done in 
the same figure. 

This cam, A5, while turned from the position shown around till 
5 comes to E, will evidently drive the follower's reacting point 
along in its path from 1 to E. The arc E5, from the construction, 
evidently equals the arc le of Fig. 208, as it should, and likewise 
for the other arcs. Hence the follower will move in its path as 
proposed, while A has the motion assigned to it. 

By Intersections. 

The cam may be drawn by the method of intersections, as in 

Fig. 210, where the follower path is 
laid oft' in the several positions in the 
inverse order of motion of A, following 
the successive equal intervals of time, 
as shown. 

For convenience in this, a templet 

may be cut to the follower path curve, 

Fig. 210 with center point at A noted. Then 

with the angles 1^42, 2^i3, etc., laid oft and noted, the several 

curves can be struck by templet. 

Now, drawing in the circle arcs from the points 1, 2, 3, etc., of 
the follower path, we obtain intersections and can draw the curve 
1, 2, 3, 4, etc. 

The method of intersections is usually employed in practice, 
and as the motion of the driver A is generally uniform, the path 
positions 1, 2 2, 3 3, etc., are uniformly distributed. 

Directional Relation and Velocity-Ratio. 

In the above cases the directional relation is constant but for 
the follower to return to its starting point, as A continues, it must 




194 PRINCIPLES OF MECHANISM. 

be changing. For this we may draw the cam in the same way, lay- 
ing off another set of points in its path for the return of the fol- 
lower, and the corresponding angles of A. 

When the angles for A are equal, and the points 1, 2, 3, etc., 
equidistant, the velocity-ratio is evidently constant, regarding the 
follower as that which moves in the path 1, 2, 3, etc. ; otherwise 
not. Sometimes the follower moves in a straight line, and some- 
times swings about a center B, the velocity of which in either case 
is to be compared with A. 

It seems unnecessary to classify cams under directional relation 
and velocity-ratio, and they will be treated here without it. 

Velocity-Ratio for a Swinging Follower. 

In Fig. 211 we have a cam A, and a follower BD, which swings 
about an axis B. The path of the point D which acts against 




Fig. 211. 

the cam is a circle arc, EF. To find the velocity-ratio between A 

and B, draw the normal DC, and then, according to Figs. 105 and 

10G, the 

, ., ■. ang. velocity of B. AC 

velocity -ratio = — , — rr — j— -. = -=5-=-, 

^ ang. velocity of A BC 

being thus in the inverse ratio of the segment of the line of centers, 
as has been found for other movements. 

When the highest point of the cam passes under D, the normal 
CD will strike at A and the angular velocity of B will be zero. 
As A moves on, the point C changes to the other side of A, and the 
motion of B will be reversed, thus changing the directional relation. 
When C is between A and B the directions of motion are opposite, 
while for C outside the directions are alike. 



CAM MOVEMENTS. 



195 



Telocity-Ratio when the Follower Moves in a Straight Line. 

When the follower point D moves in a straight line, B in effect 
is at an infinite distance away, as in Fig. 212, and BD — BC = in- 
finity. Then the linear velocity at D is to be com- 
pared with the angular velocity of A for velocity- 
ratio. 

To find the linear velocity of D, we have from 
the above the 

linear velocity of D — BC X ang. velocity of B, 
and this velocity-ratio, Fig. 212, will be 
linear velocity of D 
ang. velocity of A 

To draw an interpretation from this, take the 
expression in the form 

linear velocity of D = AC X ang. velocity of A. 
This signifies that the linear velocity of D in its 
straight path, and for the position considered, is 
equal to the velocity of the point C as if it were 
fixed upon the cam A and revolving with it. 



AC. 




Fig. 212. 



CONTINUOUSLY REVOLVING CAM, AND RETURNING 
FOLLOWER. 

By Method of Intersection. 

Assume the motion of A uniform, and that the follower swings 
about an axis B, Fig. 213, causing D to move in a circular path, 
D, 1, 2, 3, 4, forward, and 5, 6, 7, 8, D on the return. Draw circles 
through these points. There being nine divisions in the follower 
path, make also nine corresponding equal divisions in a circle 
struck through B. From each point of division as a center and 
with a radius BD strike the nine circle-arc follower-path lines 1, 
-2, 3, etc., as shown, intersecting the parallel circular lines drawn 
on the cam from A. 

Then draw in the linear cam outline through the points of in- 
tersection of these lines of like number. 

Numbers in the circles about A are in order left-handed. 
Therefore the cam is to revolve in the inverse order, right-handed. 

Here the follower moves forward and then back to the point of 
beginning in one revolution of A, and the movements may be re- 



196 



PRINCIPLES OF MECHANISM. 



peated indefinitely. The forward movement of D is faster than 
the return, there being four and five divisions to each respectively- 
These divisions of path are to represent the velocity of D* 




Fig. 213. 

Here the directional relation is changing and the velocity-ratio 
varying. 

Case of a Cylindrical Cam. 

Here A A is the elevation of the cylindric cam and the plan 
showing eight equal divisions, as in Fig. 214. At EF is the de- 




Fig. 214. 



velopment of the cylinder, showing the linear cam, the same being 
also shown on the cylinder. 

D is the point of the follower moving in a straight path parallel 
to the cylinder, so that the follower-path lines in the development 



CAM MOVEMENTS. 



197 



are straight and vertical. Above E are laid off the points 1, 2, 3, 4, 
etc., in the follower path, and through them are drawn the parallel 
cam lines. Then the linear cam line is to be drawn through inter- 
sections of lines of like number, as shown. 

By redeveloping upon the cylinder, we obtain the cylindric 
linear cam. In practice, the drawing may be made upon the cylin- 
der itself direct. 

The case of the figure is a simple one of a right-line movement 
of the follower. 

The double screw movement is an example of a double cam mo- 
tion, where a reciprocation is effected by several turns of the screw 
cam, the threads or cam grooves returning and crossing themselves. 
The follower in this case is oblong, and of length sufficient to guide 
it in one groove as it passes another. 

In case of a circular path for D, the vertical or follower-path 




rr i 1 1 1 

E 1 2 3 4 5 



Fig. 215. 

lines in the development would be circular and copied from the 
path, as shown in Fig. 215. 

It is possible for the path of D to be some other curve, in which 
case the path lines on EF should be copied from it. 

Case of a Conical Cam. 

This is similar to Fig. 214, the parallel cam lines around the 
conic surface developing in circles as in Fig. 216. The follower 
path is here taken as an element of the cone for a simple case. 
But the path of D may be a circle arc or other line when the fol- 
lower-path lines of EF will of course be drawn to the same circle 
arc or other curve as that of the path. 



198 



principles of mechanism. 



Case of a Spherical Cam. 

Here the sphere A A is to serve for the cam, as well adapted 
for the case of axes A and B at right angles and intersecting, the 




3x. ^y i 

2 

Fig. 216. 
sphere being centered at the point of intersection, as in Fig. 217. 

r-i _xn 




Fig. 217. 
The position of the follower point D is so chosen that the angle 



CAM MOVEMENTS. 199 

D" A" B" is a right angle, and the motion of D in its path will 
describe a meridian to the sphere. Then the proper number of 
meridians and parallels are to be drawn for path lines and cam 
parallels, the latter through the points 1, 2, 3, etc., of the path, as 
seen in plan at A' D' A' . The linear cam may now be drawn as 
before, through intersections of lines of like numbers. 

It is easy to imagine a less simple case, as, for instance, where 
the angle BDA is not a right angle for which a follower-path line 
of D on the sphere would not be on a meridian, but on an oblique 
great circle. 

Also, the line of B might be at a considerable distance back or 
in front of the axis AA, as for the case of axes crossing but not 
meeting, when the cam blank ADA will be in the form of a circu- 
lar spindle, or a circular zone, respectively. 

A Plane Cam or Cam Plate. 

' For this the cam will be like a sliding plate, a cam plate; as if 
the development of Fig. 214 or 215 were mounted to slide on 
guides in reciprocation, or the corresponding part of Fig. 216 to 
swing about a pivot. 

Cam with Flat-footed Follower. 

In some cam movements the follower has a flat bearing piece,. 
D, Fig. 218, instead of a point, which for the same cam changes 
the law of motion of the follower, but it presents a more extended 
bearing surface to the cam. It is evidently immaterial where the 
guide rod B is, whether to right or left, provided the pressure at 
D does net cramp B excessively. Again, D may be mounted to 
swing about a center, B' , as shown in dotted lines. 

The velocity-ratio for B is to be found as before, with respect to 
the normal DC. 

Cams for Specific Law of Motion of Follower. 

1st. A Uniform Motion of Follower, in Reciprocation in a 
Straight Line. 

In Fig. 219, take D, 1, 2, 3, 4 as equidistant points in a straight 
follower path which passes the axis A at a distance EA from it. 
Draw parallel cam circles through the points to 4, and the fol- 



200 



PRINCIPLES OF MECHANISM. 



lower-path lines as tangent to the circle EA, cutting the outer 
circle at equidistant points, 1, 2, 3, 4, etc., to 8. 





Fig. 218. 



Fig. 219. 



Then the linear cam can be traced m as shown, which, as a 
cam acting on D, will impart to it a uniform motion, to 4 and 4 
to 0, with the same constant velocity-ratio. 

The curve DBG will be an involute to the circle EA when the 
length of path is equal the half circumference of EA, and is the 
correct cam curve for the lifting cams of a miner's stamp mill, the 
stamp head rods being in the line ED. 

The velocity-ratio being constant, the normals to the linear cam 
curve DHG, at any contact of D, will all intersect the line EA 
produced m one and the same point E at the left of A. Also, the 
normals to the cam curve DIG will intersect the line EA at the 
same distance from A on the opposite side. 

If the construction be such that the point E coincides with A, 
the linear cam curves become Archimedean spirals, and the cam 
one of constant diameter, and known as the heart cam. 



2d. Law of Motion that of the Crank and Pitman. 

In Fig. 220, A is the center of the cam, the center of the 
crank shaft, and abode half orbit of the crank pin, in which the 



CAM MOVEMENTS. 



201 



latter is supposed to move at a uniform rate, making the spaces ab> 
be, etc., in equal times. 

The pitman has the length ao, bD, etc., e4, so that the spaces 1, 
1 2, etc., are passed in equal times by the end D of the pitman ID, 
:and with a varying movement in accordance with the so-called 
•crank and pitman motion. Through these points draw the par- 




Fig. 220. 

allel cam circles and divide them into equal parts by the follower- 
path lines as tangents to the circle A C and to equal points of 
•division, 0, 1, 2, 3, etc. 

Then draw in the linear cam, through points of intersection of 
path and cam circle lines of like numbers. This cam will give the 
follower D the same motion as would the crank and pitman; the 
orank or the cam both being supposed to revolve with uniform 
motion. 



3d. Law of Motion that of a Falling Body. 

In Fig. 221, A is the center of the cam, D the follower to move 
in the path D4, its velocity in the first half of which path is to be 
accelerated as that of a falling body, and in the last half to be 
retarded according to the same law. 

To express this law on the line D4 of the path, draw a parallel 
ag, on which as an axis draw a parabola abc, and an equal one, 
gdc, intersecting it at c. Make several equal divisions gf,fe, etc., 
and from these point off equal divisions, erect ordinates ec,fdb, etc., 
and from the points of intersections, d, c, b, etc., draw horizontal 
lines over to the path B4. Then if the spaces flj,ji, etc., or 1, 



202 



PRINCIPLES OF MECHANISM. 



1 2, etc., down to the middle of ag are described in equal times^ 

the law of motion is that of a fall- 
ing body, from the known prop- 
erty of the parabola. The re- 
versed parabola gdc will give the 
desired points in retardation. 

One peculiar property of the 
action here is that the inertia 
of the parts of the follower will 
cause a constant resistance in the 
follower path — a probable advan- 
tage where the follower is very 
heavy. 

In practice, numerous lines 
should be used in the construc- 
tion, a few only being employed 
here for clearness of figure. 




Fig. 221. 



4th. Tarrying Points for Follower. 

It often happens that the follower is to make a movement, then 
halt momentarily, then make another movement, then halt, etc. 
These halting or tarrying points are easily pro- 
vided for in the cam by introducing circular 
segments, as shown in Fig. 223. 



5th. Uniform Reciprocating Motion. 

The so-called heart cam is a common device 
for this, an example of which is shown in Fig. 
222. As a linear cam it has a constant diameter 
through the axis, and when the follower is located 
to coincide with this axis and is provided with a 
bearing point against the cam, both above and below 
will be no backlash between the cam and follower. 




Fm. 222. 

it, there 



6th. The Heddle Cam. 



This can be used to work the heddles or harness in looms where 
the intervals of motion and rest are about equal. Fig. 223 illustrates,, 
by photo-process copy, a working cam of this kind. 



CAM MOVEMENTS. 



203 



To make a heddle cam of constant diameter, uniform motion, 
and with tarrying intervals of time equaling the motion intervals : 

In Fig. 224 divide the cam into four equal parts, retaining the 
opposite segments DH and EG for the tarrying intervals. The 
remaining opposite segments must 
serve for the linear cam lines. - In 
these segments draw equidistant cir- 
cle arcs, and also radial lines, in equal 
number. The linear cam lines may 
then be drawn in through points of 
intersection as shown, giving us the 
linear cam outline DbEGeHD. 





Fig. 223. 



Fig. 224. 



All lines of this cam through A will have the same length, or 
the cam will have a constant diameter, and it will just fill the space 
between follower points D, G on a slide BB'. 

The curves DIE and GeH are Archimedean spirals. 

7th. Easements on Cams. 



The cam of Fig. 224 will instantaneously start the follower into 
full motion, the latter remaining uniform until stopped with equal 
abruptness. This is objectionable in any case, and when the fol- 
lower is heavy, it is destructive. This is obviated by what may be 
termed easement curves for the cam, or easements. 

An Arbitrary Easement may be drawn, as in Fig. 225, where A 
is the center of the cam, and G a point of abrupt change between 
the tarrying and cam arcs. 

Assume equal angles EAG and GAF, and divide them into 



204 



PRINCIPLES OF MECHANISM. 




Fig. 225. 



equal parts by lines extended outward. Then note points a, b, c, 

etc., over to the line through G, and in 
increasing distances from EG, such as 
estimated to give a proper easement curve 
when drawn through the points. Lay off 
e the distances fa, gb, etc., on ie, hd, etc., 
until the center radius Gc is reached. 
Then draw the easement curve EabcdeF 
through the several points. 

Treating the cam, Fig. 224, in this way, 
filling in at E and G, and taking off like 
amounts, on like radii and angles at D 
and H, we modify the cam, Fig. 224, by 
supplying easements, and without altering the constancy of its 
various diameters. 

As thus treated, however, we encroach upon the tarrying arcs 
and shorten the times of rest of the follower. We may, however, 
provide for the angle EG in laying out the cam angles. 

8th. To Confine the Easements within the Assigned Cam Sectors, 
take DAH, Fig. 226, as the allotted sector for the linear cam arc 
sought, including the easements. Let it be assumed that the ease- 
ments shall start the follower as by the law of the crank and pitman 
motion; then, when full motion is at- p 

tained, to continue that motion of fol- 
lower uniform till approaching the 
arc of rest, and then to stop the fol- 
lower by the same law of the crank, q 

To this end draw the quarter cir- 
cles DI and GF from centers K and J* 
on the follower path, and connect G 
and / by a common tangent. Then 
with dividers note points in equal di- 
vision from D through /and G to F. 
Project these points to the path line 
DF, and draw circle arcs about A Fig. 

through the sector. Also draw an equal number of radial lines 
equidistant between D and H. The linear cam line DLE can now 
be drawn in, which will be found to have easements at D and E 
which are tangent to the arcs of rest for follower, and without en- 
croachment thereon. 

This linear cam will be found steeper at L than the cam of Fig. 




CAM MOVEMENTS. 



205 



224, and it may be a question which cam is preferable. But by 
making the circle arcs DI and GF smaller the easements may be 
more limited, and the inclination of the central portion of the cam 
arc less severe. 

Other laws of easement curve may be adopted, as, for instance, 
DF may be taken as the base of a cycloid, in which the same num- 
ber of divisions may be made as in DIGF. If that arc be shorter 
than the one here drawn, the resulting linear cam will have a milder 
declivity at L. 

The use of any curve DIGF, which is symmetrical with respect 
to a perpendicular to the middle point of DF, will give a linear 
cam line DLE, which, if paired with a like one in the opposite 
quadrant, as in Fig. 224, will constitute a cam of constant diameter 
through A. 

In Fig. 226 the law of motion of follower is represented by the 
spacing along the follower path DF. 

Case of Flat-footed Followers. 

Let the law be that of a crank and pitman motion ; determine a 
cam of the kind shown in Fig. 218. Draw the circle of the crank 




I 



□> 




Fig. 227. 



Fig. 228. 



at DIE, Fig. 227, and project points a, i, c, etc., of division into 
equal crank arcs, to the radial line DF, giving points d, e,f, etc. 



206 



PRINCIPLES OF MECHANISM. 



Draw the same number of diametric lines through A as FA, 
GA, etc., to represent the motion of A. Then draw cam parallel 
circles from points d, e, f, etc. At intersections, as at F, G, etc., draw 
the follower line of DM as representing the position of DM relative 
to A when FA, GA, etc., have revolved to the line AD. That is, 
if DM is perpendicular to AD, draw FJ, GL, etc., perpendicular 
to the lines A F, A G, etc. 

Tangent to all these position lines of MD draw in the linear cam 
curve, as HLJD. For this case the half may be copied on the other 
side of HAD as dotted, giving the same law of return of follower. 

For this particular case and law, the cam is a circle or crank pin. 

In cams with a follower consisting of a straight footpiece D, 
where the law on the return movement is the same as for the for- 
ward, the cam will be one of constant breadth as reckoned on a 
normal to the linear cam; and the follower may be made as a rack 
or frame DE, Fig. 228, with parallel sides. 

Thus cams which while revolving just fill the space between a 
pair of points fixed upon the follower, as in Figs. 222 to 224, and 
229, may be called cams of constant diameter; while those likewise 
just filling the space between parallel lines, or footpieces fixed 

upon the follower, as in Fig. 228, 
may be called cams of constant breadth. 

Cams of Constant Diameter and 
Breadth. 

1st. Constant Diameter. 

Besides those mentioned above, 
Figs. 222, 223, 224, 225, and 226, there 
may be cases where a specific law of 
motion is to be followed for a half 
revolution of cam, while the motion 
for the remaining half is immaterial 
as to law. 

For this, as before, the follower 
points DK, Fig. 229, must move in a 
straight line through the center A. 

Taking DEGIK, Fig. 229, as the 

essential cam for the half revolution, 

draw a series of lines, EF, GH, LJ, etc., through A, and make them 

all equal in length to DK, noting points FHJ, etc. Then the 

required counter half of the cam will be the dotted line KFHJD. 




Fig. 229. 



CAM MOVEMENTS. 



207 



2d. Cams of Constant Breadth between parallel lines are possible, 
as in Fig. 230, where a half of A is assumed, and the forked 




Fig. 230. 

piece B to swing around the axis will just span the half cam 
at DE. Then by placing the fork in various positions as at F 
and G and drawing a line at G, again at H and I and drawing a 
line at i, etc., for a sufficient number of positions, and drawing an 
enveloping curve EGID, we have the remaining half of a cam, the 
breadth of which for all positions just fills the fork-shaped follower. 

The outline of the cam, also its law of motion correspondingly, 
are limited, however, by the circumstance that the center of cur- 
vature of every part of the outline must be within that outline. 

3d. A Cam of Constant Breadth as between parallel lines may 
be drawn by aid of a series of intersecting lines, as in Fig. 231. 

Draw any system of intersecting straight lines. Then, beginning 
at some remote intersection as at a, draw a circle arc A from line 
to line, as those intersecting at a. This arc will be normal to the 
lines limiting it. Next, seeing that the 
arc B will meet the line intersecting at 
b, take b as a center and draw the arc B. 
The next arc will be G drawn from the 
center c, so that C will be normal to 
both lines intersecting at c. Draw the 
arc D from the center d ; the arc E from 
e; the arc F from/; the arc G from g; 
the arc i^from h; thus returning to the 
place of beginning. Arc If closes upon 
arc A exactly, if the work is right. 

A little consideration will show that this cam, for every position, 
will just fill between a pair of parallel lines at a distance AE apart. 




Fig. 231. 



208 



PRINCIPLES OF MECHANISM. 



Also, it will just revolve, with a close fit, in a square hole of width 
AE each way; or in a rhomboidal hole of width AE each way, and 
with any angle between the parallel sides. Also, the hole may be 
bounded by parallel circle arcs, instead of parallel straight lines, 
partly or entirely, as in Fig. 237. 

Cams with Several Followers. 

1st. One Cam may have Two or More Followers, as shown in Fig. 
232, the velocity-ratio of which may or may not be the same, as 
depending upon the construction of followers. 




Fig. 232. 



2d. The Effect of Two Followers may be obtained by combining- 
two followers into one, as in Fig. 233, where the combined parts 
DG are solid, with a slide branch, E, the latter working in a box on 




Fig. 233. 
a second sliding piece, F, so that motion may be taken off at B and 
J in two directions, 



CAM MOVEMENTS. 



209 



3d. Fig. 234 gives a Similar Construction, in which DG swings 
about a pivot E, and that in turn about a second but fixed pivot F. 

The follower connectors B and J may take the motion in two 
directions. One spring may serve to return the compound follower 
for the two directions of movement. 




Fig. 234. 



4th. Cams of Constant Breadth may return the compound fol- 
lower positively, as in Fig. 235, where, as stated at Fig. 231, the 
cam of constant breadth will just fit and revolve in a square opening. 




Fig. 235. 

5th. The Four-motion Cam is shown in Fig. 236, and constructed 
by aid of intersecting lines in the manner shown in Fig. 231. 

When the lines ^4i^and AE are at right angles and EF inter- 
sects them at equal angles of 45 degrees, and the cam is drawn from 
centers AEF, as shown, the follower!), when pivoted at. a rela- 



210 



PKINCIPLES OF MECHANISM. 



tively great distance to the right, will have every point moving in 
an exact square; as, for example, the point d will describe the 



/C\ 


f\ 


^ 




1 /v> 


A 1 


D 


L ^ 


V 


v 


>&-a 






-'a S 
e f 


U (J 

O O 



Fig. 236. 
square defg, where de equals the difference of the construction 
radii a and b. 

This cam has been used for sewing-machine feeds, for which it 
is well adapted, except that for this, one side of the follower should 
be cut out and an adjustable stop put to it to vary the feed. 

In drawing the cam, it is immaterial as to motion how large it 
is, provided the periphery goes through or outside the points E and 
F, outside being preferable to avoid sharp intersections at those 
points. 

In Fig. 237 is illustrated a working cam of this kind, where two 




Fig. 237. 

of the parallel lines of the follower are straight, and two are par- 
allel circle arcs struck from the upper joint pin. This cam fits 



CAM MOVEMENTS. 



211 



closely both ways in the opening, and the circle arc sides have the 
effect to keep the vertical connec- 
tor bar quiet for two of the four 
movements, its motion being inter- 
mittent in reciprocation. 

6th. The Peculiar Shaped Cam, 
called a Duangle by Reuleaux, is 
shown in Fig. 238. The duangle 
closely fits in the triangular open- 
ing in the follower for the entire 
revolution. 

The follower moves forward and 
immediately back, and there tarries 
nearly stationary for about a sixth 
of a revolution, when another reciprocation is commenced. 

An interesting double cam motion, which Willis would class as 
resulting in an aggregate path, is given in Fig. 239. Two arms 
work agreeably to the end of carrying a pencil point in such a 
manner that it writes the letters " O.S U." and places the periods. 
Model due to the ambitious energy of D. F. Graham. 




Fig. 238. 




Fig. 239. 



Return of the Follower. 

In most cases in practical machinery it is very essential that the 
return of the follower be positive in order to avoid breakages, 
though sometimes gravity or a spring may be employed where the 
velocity is not unduly high, and damage not likely to occur. 

That the Action of Gravity may Quicken the Return of the end 
D, put the weight W near B, Fig. 240, and make the arm D W com- 
paratively light, so that as W falls by gravity the end D will move 
more rapidly than W by reason of the leverage. 

In this case take g, the " center of percussion " of the piece BD. 



212 PRINCIPLES OF MECHANISM. 

When g is in a horizontal line through B, g starts to fall just as 




Fig. 240. 
last as a free weight, and the end D faster in proportion to its 
greater distance from the pivot B. 

A Positive Return is readily obtained by making a groove for the end 




Fig. 241. 
D to run in, so that it is compelled to move both ways, as in Fig. 24U 
A positive return may also be insured by means of an extra cam, 




Fig. 242. 
as shown in Fig. 242, one cam working in unison with the other 
to prevent backlash between either follower and its cam. 



CAM MOVEMENTS. 



213 



In certain cases the cam may be made of constant diameter 
with the follower moving in a straight line, as shown in Fig. 243, 
when the follower points DD may just contain the diameter with- 
out backlash whereby the follower is compelled to move both ways. 
This may serve where a certain law of motion of follower is to be 
insisted upon for only a half revolution of A ; 
or the same law for the forward and return 
motion of follower as in Figs. 222 to 226. 

In cams of constant breadth the return of 
the follower may be assured, as in Figs. 228, 
230, 236, and 237. 

To Relieve Friction. 

Where the follower is compelled to drag 
over the full extent of the periphery of the cam 
at each revolution extensive wear is very likely 
to occur. Lubrication will help, but it is diffi- 
cult to maintain this on surfaces so exposed to 
air and to the tendency of centrifugal force 
to throw the oil off. The usual remedy is to 
put a roller at D, as in Fig. 244, which can 
track along on the exposed edge of cam, regardless of lubrication. 

While rubbing of surfaces occurs at the pin there will be sur- 
face bearing, instead of a line bearing of a fixed terminal D on the 





Fig. 244. 



edge of A. Also there will be more favorable conditions of lubri- 
cation between roller and pin than between terminal and cam. The 
amount of rubbing action of surfaces will also be reduced in the 
ratio of the diameter of roller to diameter of pin. 

In conical cams the roller is sometimes made conical. 



214 



PRINCIPLES OF MECHANISM. 



In stamp mills a disk roller, Fig. 245, is used, which, though 
largely reducing the friction, does not do so as completely as the 
cylindric roller. Here A is the cam lifting the 
rod EF by acting against the collar D on the 
rod. The rod and collar both turn together as 
A may dictate. But owing to the thickness of 
A, with D rotating, the radius of D at one side 
of A differs from that of the other side, so that 
there is a combined rolling and torsional twist 
between the surfaces, and not an entire relief 
from slipping. 

A flat-foot follower, as in Figs. 227 or 230> 
will be found better for endurance in wear than 
a sharp edge, as in Fig. 243. 

Modification of Cam Required by 
Follower. 

The immediate result of solution of a cam 
motion is usually a so-called linear cam, from 
which the practical cam is to be obtained. 
In Fig. 246 the linear or theoretical cam is the curve Dabc, 
etc., which is the line as traced on A, which the theoretical fol- 
lower point D is to follow as A revolves. 

To Introduce a Follower Roller at D and maintain the theoretical 
action, the center point of the roller must follow the linear cam 





Fig. 246. 



Dabc, To insure this, strike circles equalling the roller in diameter 
at numerous points along the linear cam, as at a, b, c, etc. Then 
draw the practical cam lines tangent to these circles throughout. 



CAM MOVEMENTS. 



215 



If these enveloping cam curves are drawn at both sides of these 
circles, we have a drawing of the cam groove for cam A, providing 
for a positive return of the cam follower. 

In practice, this groove is best executed in metal, by using a 
cutter of the same shape and diameter as the roller D, and so 
mounting it in a cutting machine that the cutter while cutting is 
compelled to move with its center following the line abc, etc. 

In Cams with Salient Angles, as in Fig. 219 and the heart cam of 
Fig. 222, a sacrifice must be made from the theoretical action of 
the cam by the introduction of 
a roller, as shown in Fig. 247. 
Let FNEMG represent the lin- 
ear or theoretical cam and the 
line the center of the roller 
should follow. Drawing a series 
of roller circles, and the practi- 
cal cam HIJ tangent to the cir- 
cles, it is found that at / the 
cam is foreshortened by the 
amount LE. That is, for the 
practical cam HI to compel the 
roller center to move on the 
linear cam FN to E, the cam 
surface HI would require to be 

extended to some point near M, where its normal would strike E. 
But as the cam is cut away beyond I by IJ, the roller is at liberty 

to swing around its point of con- 
tact with 7, the center of the 
roller describing the circle NLM 
and failing to reach the point E 
in the linear cam by the amount 
LE. 

This circumstance may at 

times prohibit the use of the 

antifriction roller, as depending 

upon how essential the shortage 

EL may be. Or, at times, the 

roller may be used of diameter 

Fig. 248. smaller than otherwise preferred. 

For a Thickened and Rounded Follower Extremity, D, as in Fig. 

248, where some point D is to follow the linear cam as shown, cut 




Fig. 247. 




216 



PRINCIPLES OF MECHANISM. 



a templet BD of the right shape at D for the follower, and with 
the point D notched by a V notch as shown, through which 
the linear cam can be seen. Then, with a circle about A 
through B, place the templet at numerous positions, as at B'D', 
and draw a curve at the end, as shown. A cam line, traced tangent 
to all these curves as shown, will be the practical cam required. 

For an Edge Cam, as in Figs. 242 and 244, and for a disk or face 
cam, with a cam groove in the side, as in Fig. 246, the roller should 
be cyhndric without question. 

The Best Form of Roller has been a matter of some question, as 
some designers use a cylindric and others a conic one for a cylin- 
dric cam, such as shown in Figs. 214 and 215. 

With regard to the action on the outside surface of the roller 
alone, it appears that when the roller is moving longitudinally in 




Fig. 249. 



its groove, if such might be, its form should be the cylinder; while 
when the cam is revolving and the roller nearly stationary its form 
should be conic, with the vertex of the cone at the axis of the cam. 
For forms of the cam between the above limits the roller would 
seem to require some compromise form as between the above cylin- 
der and cone. A little consideration will show, however, that a 
perfect form for simple rolling contact in this case, of axes of cam 
and roller meeting, does not exist. 

Suppose the cam approaches one of the same velocity-ratio for- 
ward and back, as in Fig. 249, where A A' is the cylindric cam and 
EF its development. Take ab for one side of the cam groove, 
and cle the same in the development. 

Now, if the roller is made conical with its vertex at the axis A, 
when the roller rolls from a to b at the surface of the cam it 



CAM MOVEMENTS. 217 

would be necessary for the vertex to roll from a to c on the axis A, 
which is clearly impossible since the vertex is a point. From this 
it would seem that the roller must have some size at the axis; and 
probably the best that can be done by approximating it is to lay off 
the distance ac atfg, where fd equals the radius of the cylinder A; 
draw a?/ and eg produced to meet in 0, and take dO as the length 
of. the cone from the large end of which to cut the follower roller. 

Perfect rolling of the follower roller rolling along a cam surface 
ab would require that the axes of cam and roller pass each other 
with a distance between, and that the roller be a hyperboloid of 
revolution; also, that the distance between the axes varies as the in- 
clination ab of the cam curve varies. It therefore seems impossible 
to obtain a perfect follower roller for a cylindric cam, that is, one 
where the outside surface of the roller simply rolls on the surface 
o± the cam groove, because the action will of necessity be partly 
rolling and partly torsional slip of surfaces, and vary with slope of ab. 

This torsional slip of surfaces will be the same for a truncated 
conic roller with vertex at axis o± cam and moving in a cam groove 
parallel to the axis, as for a cylindric roller of equal length moving 
in a groove encircling the cam; and according to Fig. 249 these 
torsional slips tor both rollers will be alike for a groove at an angle 
of 30 degrees with the circles of the cam surface. 

The Action of the Roller upon the Pin and its Shoulders is of 
importance, as it is easily seen that the conic roller will be severely 
pressed against its shoulder, causing friction on a surface of larger 
diameter than the pin, thus introducing a very serious resistance to 
rotation of roller — an objection which, probably, by far outweighs 
the advantage of a conic roller, except in cases where the cam is of 
large diameter relative to the throw. 

With regard to the pin for supporting the roller: when it can 
be made conical, with the same angle of convergence as the conic 
roller itself, the end thrust causing shoulder friction will be mostly 
avoided, and the conic roller will be much more acceptable. One 
drawback here, however, may be found in the large average size of 
pin, and it may be advisable to make it part way cylindric and the 
remainder conic. 

For a Conic Cam the question of best form of follower roller is 
still more involved, and it is probably advisable to adopt the cylin- 
dric one. 

For the Spherical Cam there seems to be no question but that 
the roller should be conical, since here the vertex of the cone can 



218 PRINCIPLES OF MECHANISM. 

remain at the exact center of the sphere, and there will be theoreti- 
cally perfect rolling between the roller and its cam groove. But 
even here the shoulder friction due to the endlong thrust will be 
found a serious objection to the conic form of roller unless the 
axial pin for the roller can also be conical to match, or partly conic 
and partly cylindric. 




CHAPTER XIX. 
INVERSE CAMS AND COUPLINGS. 

I. The Inverse Cam. 

This term may be given to a movement which has the elements 
of a grooved cam and follower, but where the driver has the pin or 
roller and where the follower has the groove; styled by Willis the 
pin and slit. 

The inversion of the movement is to 
avoid dead points that would in some 
cases occur when used as a cam move- 
ment. 

The peculiarity which distinguishes 
this from the cam movement consists 
in the fact that here the pin recipro- 
cates in the slot or groove, while in the 
cam it does not. 

The slotted piece B, Fig. 250, is the Fig. 250. 

follower and A the driver with directional relation constant. The 
velocity-ratio as in cams is 

ang. velocity of A _BO 
ang. velocity of B~~ AC 

The slot may be made straight on a radial line or not, and a 
block may be fitted on the pin and in the slot to avoid wear by 
extending the bearing surface. Thus equipped with the slot radial, 
this movement is sometimes known as the Whiteworth Quick Re- 
turn. It is considerably used on English shaping machines. 

An example of this movement for directional relation changing 
is given in Eig. 251. In this there is a point of tarrying of the 
driven piece by reason of the slot having quite a portion made to 
the circular path of the driver. Thus by curving the slot, modi- 
fications of motion may be obtained. 

This movement has been used to give motion to the needle-bar of 
a sewing-machine. 

219 



220 



PRINCIPLES OF MECHANISM. 



Another example is given in Fig. 252 of a needle-bar cam 
motion much used in sewing-machines. It corresponds somewhat 
with Fig. 251, except that the axis B is 
in effect removed to infinity by the 
mounting of EF on a sliding bar. 

At F the slot may be made to cor- 
respond with the circle arc described 
by the pin; so that the needle will 





Fig. 251. 



Fig. 252. 



stand stationary while the shuttle passes the loop of thread. To 
form the initial loop, a quirk may be introduced in the slot at E. 

In some applications in heavy machinery, this slotted piece EF 
has a straight slot and a block on the pin fitted to slide in the slot. 
Thus the pitman has been avoided in steam-engines and the engine 
correspondingly shortened. 

By a sufficiently wide slot in FE, the movement may be placed 

at an intermediate point on a straight 
shaft and eccentric. 

II. Couplings by Sliding Contact. 

Oldham's Movement with direc- 
tional relation and velocity-ratio 
constant is illustrated in elementary 
form in Fig. 253 for connecting 
axes that are parallel but not co- 
incident, acting by sliding contact. 

The velocity-ratio is constantly 
equal to 1, as easily seen in the small 
diagram of a section normal to the shafts. The arms of the con- 
necting cross which slide in the sockets made fast upon the shafts 




Fig. 253. 



INVERSE CAMS AND COUPLINGS. 



221 



must constantly pass through both of the axial points A and B; 
and as they form a right angle, the point of intersection of the 
cross must follow the circle of diameter AB as shown, since all 
lines at right angles drawn from the extremities of a diameter 
meet in the circle to that diameter. The extent of sliding per 
revolution on each branch of the cross equals two diameters, AB. 

If the cross is not right-angled, the same is true of the angular 
velocity, as shown in Fig. 254 ' but the 
amount of sliding is greater, since the 
circle is thrown to one side and increased 
in diameter. 

A Complication of Movement results 
from placing the axes out of parallel, as in 
Fig 255, so that they meet at some point, 
0. Then ED shows one position of the 
cross, the angle being at E and F. An- 
other position, a quarter-turn away, is 





Fig. 254. 



Fig. 255. 



shown at HI, with the angle at / and J. These diagrams, com- 
pared, show that the shaft B has endlong motion to the extent 
G H, twice in a revolution, regarding A as without end play. The 
angle point of the cross, however, always remains on the line 1G, 
or plane normal to A, and describes a circle KLM in that plane. 

To study the relative motions of the shafts A and B, take the 
cross as right-angled and in an intermediate position, KL, LM. 
The A -branch will always be found in the plane PN, normal to A, 
and the ^-branch in a plane NQ, normal to B. These planes will 
intersect in a line perpendicular to the plane of the axes A and B, 
or in some line X, RL, for the position of the cross as shown. 
Draw a circle about L on the plane normal to A, and an ellipse to 
the same center, to represent a circle on the plane normal to B. 
Taking KL for the A -branch of the cross, and, as in the plane of 
the paper, the 2?-branch in its own plane will appear at ML with 



222 



PRINCIPLES OF MECHANISM. 



MLK a right angle because lines at right angles in space will 
appear at right angles in projection, when one of the lines is paral- 
lel to the paper. Hence, if we move B from parallel to A into its 




Fig. 256. 

inclined position, while A and KL remain fixed, the point b in the 
circle must move to c in the ellipse. To determine the angular dis- 
turbance of B in this movement, swing B and its point, c in the 
ellipse, back, without angular disturbance, to parallel with A, when 
the ellipse returns to the circle and the point c will fall at d, cd 
being perpendicular to RL. 

Then cLd will be the change in the angular position of the 
i?-arm of the cross as due to the swinging of B from parallel to A, 
to the position BO. 

Let x and y represent the angles SLd and SLb respectively 
and a the angle between the axes A and B. Then Fig. 256 will 
show the relation of these angles, from which we get EN ' = 
DN cos a, EN = LS tan y, DN = LS tan x, which gives us the 
relation 

tan y 



cos a = 



tan x 9 



in which, a being constant, y may be found when x is given, or x 
found when y for any point in the revolution is given. 

It may be noted that the same figure in cross-section at RKLM 
is obtained whether the intersection is at or at P; also the same 
equation. But it is clear that when is at P the movement re- 
duces to that of the Hooke's universal joint, which is free from the 
sliding motion of Fig. 255, and hence the latter can claim no ad- 
vantages over the Hooke's joint. 

The velocity-ratio could be found by calculating a series of 



INVERSE CAMS AND COUPLINGS. 223 

angular positions from the formula and comparing them. But 
this is best done by differentiating the formula and obtaining 

, . , , . ang. veloc. A dx cos 2 x 

velocity-ratio = — - -. ^ = -r- ■ = — -$— sec a 

J ang. veloc. B ay cos y 

_ 1 — sin 2 x sin 2 a 

cos a 

cos a 



cos' 2 y sin 2 a 



When a equals the angular velocities are equal, as they evi- 
dently should be, and the velocity-ratio = 1. 

These equations are the same as given by Willis, p. 452, for 

Hooke's joint. 

dx — V2 

For a = 45 degrees; -j- = 4^2, or = -— , for maximum and 

minimum values as occurring f or x = and 90 degrees respectively. 

dx 
The velocity-ratio is t-= 1 for tan 2 y — cos a = cot 2 x. These 

equations are the same as found by Willis and Poncelet for the 
Hookers joint. 

A Peculiar Movement transmitting motion from A to B is illus- 
trated in Fig. 257, in which the contact between the parts is by 
sliding, except where the axes are in direct line. In the model, 
the axes may be arranged parallel or at various angles and at var- 




Fig. 257. 

ious offsets, as in Figs. 253 and 255, and it appears to be one way 
oi realizing those cases in material form, except that here the axis 
B is not compelled to slide endwise. 

To study the velocity-ratio it is most convenient to imagine the 
intermediate slotted piece to be replaced by a cross the branches 



224 



PRINCIPLES OF MECHANISM. 



of which are perpendicular to the slots they stand for. The move- 
ment will then serve for any and all the conditions brought out in 
Fig. 255 or 253 and with the same law of velocity-ratio. 

These joints all have two points of maximum and two points of 
minimum velocity in each revolution, as also a pair of the two-lobed 
elliptic wheels of Fig. 44, but the law of velocity-ratio of the latter 
is different. 




Fig. 258. 

When the movement of Fig. 257 is set with axes parallel, it acts 
the same as the three disks in Fig. 258, which latter movement 
was used by Oldham in appliances employed in the Banks of Eng- 
land, by Winan in his Cigar Boat, and also by 0. T. Porter to couple- 
the shafts of an engine and dynamo nominally in line, but practic- 
ally a little " off line " by reason of temperature, flexure, etc. 



CHAPTEK XX. 



ESCAPEMENTS. 

DIRECTIONAL RELATION CHANGING. VELOCITY-RATIO 
CONSTANT OR VARYING. 

An escapement is a movement in which the follower is driven a, 
distance, usually by sliding contact, to where the driver is allowed 
to pass free for a little space, when another engagement by sliding 
contact occurs to drive the follower back to a position for repeating 
an engagement like the first. 

Power Escapements. 

Fig. 259 illustrates an escapement of a design suitable for use in 
heavy machinery in which the motion is continuous and where there 




Fig. 259. 
is no lock such as used in most cases of escapements for clocks and 
watches. 

As F escapes from E, the arm G should be near to D to prompt- 
ly engage; and the point of initial contact at D should be as near 
to the longitudinal line through A as possible, to relieve the blow 
due to initial contact. 

Escapements are mostly employed in timepieces, and should be 
as nearly as possible such as to give to the vibrating pendulum or 
balance equal impulses at all times, and be as free as possible from 
hindering the vibration by frictional contact of parts with the pen- 
dulum or balance. That is, the higher essentials for fine time- 
keeping are, 1st, an isodynamic or equal-impulse escapement; 2d, 
an isochronous pendulum or balance; and 3d, freedom of vibrating 

225 



226 



PRINCIPLES OF MECHANISM. 



parts from contact with the other pieces, except when receiving the 
impulses. There are other considerations, such as temperature, 
position, etc., which are outside of our present topic. 

The Anchor Escapement. 

In Fig. 260 is shown a so-called anchor escapement, the name 
being due to the resemblance of the vibrating piece to the ship's 
anchor. 

As shown in full lines, we have the dead-beat escapement, in 
which the escape-wheel A stands still while a tooth rests on a pallet, 
notwithstanding the movement of the pallet. As shown, the tooth 
1 is about to move forward upon the pallet HG, as the latter ad- 




Fig. 260. 

vances left-handed. Reaching IT, a further movement of H to 
the right and return allows / to remain stationary, because the 
pallet from H back is formed to a circle arc about the center B. 
A like action occurs when the tooth rests and slides upon the pallet 
E. The teeth of A should be so formed and cut away as to permit 
the pallet to move a considerable distance after the escapement of 
a tooth and before the return of the pallet occurs, so that the pen- 
dulum may complete its swing. 



ESCAPEMENTS. 227 

Thus the tooth is just on the point of escaping at D, following 
which / moves forward upon the pallet H as the latter swings to the 
right, allowance being made for HG to move still farther toward A. 
As HG returns, the pressure of the tooth 1 upon the slope HG im- 
parts an impulse to GHB toward the left. Similarly at D, the tooth 
is just completing its impulse on ED toward the right. These im- 
pulses overcome the retarding influences of the air and other re- 
sistances acting upon the pendulum, thus maintaining its motion. 

To construct this movement, the pallets are somewhat thinner 
than the half of the pitch of the escape-wheel, so as to give a slight 
drop 1H to insure the landing of / upon H at a slight distance from 
the bevel HG. From H and G back, the pallets are formed of cir- 
cle arcs struck from the center B. The bevel HG may be as- 
sumed by a line drawn to J tangent to a circle struck from B. The 
pallet ED should be formed to the same circles as HG, and beveled 
by a tangent to the same circle JK. Then the angle DBE will 
equal the angle HBG, as it evidently should to balance the im- 
pulses. 

It has been proposed to put the impulse bevel upon the teeth of 
the escape-wheel instead of the pallets. Fig. 260 would nearly fit 
the case by turning A backwards when, as the tooth of D escapes, 
the tooth MN would fall upon the circle of the pallet G as an arc 
of repose, until, when the pallet returns, it would receive an im- 
pulse in sliding along the bevel NM until, when M escapes at G, 
of the next tooth OP will land upon the pallet D, and in due time 
impart an impulse from the bevel OP. 

Again, the impulse bevel may be divided between the pallet and 
the teeth of the escape-wheel, as really done in the ordinary " lever 
escapement " of watches. 

The recoil escapement is obtained from the above by producing 
the bevel lines HG and DE, as shown in dotted lines, and modify- 
ing the teeth of the wheel to some shape as dotted at I. Then, as 
the tooth at D escapes from DEF, the tooth / will strike upon the 
bevel GHL, and, as the pendulum moves still farther in the same 
direction before returning, /will be forced to slide up towards L, 
thus giving to the escape-wheel a slight backward movement, called 
the recoil. 

The recoil escapement is the most common one in ordinary man- 
tel clocks, and regarded as inferior to the dead-beat, which latter is 
usually introduced into the finer mantel clocks, regulators, astro- 
nomical and many tower clocks. 



228 



PRINCIPLES OF MECHANISM. 



The Pin-wheel Escapement. 

This is so named because the escape-wheel has pin teeth, and it 
is shown in Fig. 261. 

The pallets are formed to circle arcs ej,fi 9 gl, and hk, terminated 
with bevels ef and gh upon which the pins slide to impart the im- 
pulses to the pallets. A pin of the escape-wheel is shown as resting 
upon the pallet hk. If the pallets are moving toward the left, the 
pin slides upon the circle arc, or " arc of repose," toward k until the 




Fig. 261. 

pendulum reaches its limit of swing, when it returns, and also the 
pallet, which, on arriving at the pin, permits the latter to slide 
down the bevel hg, imparting the impulse. It escapes from the 
pallet ghk and alights upon fi, of the pallet eft. The pendulum 
completes its movement and returns, when the pin slides down 
fe and imparts a second impulse opposite to the first. As the pin 
escapes at e, the next pin drops upon hk, to repeat the movements 
of the former one, etc. 

To construct the movement, draw circles from the center B 
for the pallets, the latter having thicknesses such that the two, 
with a pin between, will swing between two adjacent pins of the 
escape-wheel, with a necessary slight clearance. Then the bevels 
ef and gh are so determined as to subtend the same arbitrary angle 
at B as hBc, besides allowing a small safety angle ab equal to cd, as 
providing for the distance from the impulse bevel back to the land- 
ing point of the pin upon the pallets. This angle may be small 
and possibly zero for cylindrical pins. When the pin escapes at e 



ESCAPEMENTS. 229 

(see dotted line), the next pin should land upon hh at the allowed 
angle aBb from h. Also, when the pin escapes from lig (see dotted 
line struck from D with a radius equal to distance from A to inside 
of pin) it should drop upon/t at the same angle cBd equal to aBb, 
from the initial point /of the bevel. 

This escapement has been considerably employed for tower 
clocks, and has the advantage that the pins may conveniently be 
hardened, or even made of glass rod or cut jewel stones. In some 
cases the upper half of the pins are cut away, since this portion is 
not acted upon by the pallets, thus permitting the placing of adja- 
cent pins a whole pin diameter closer. Then with the same num- 
ber of pins and thickness of pallets, the wheel A will be made 
smaller, and the strength of an impulse will be materially increased 
other things the same. Also, the drop of a pin on escaping will be 
reduced by nearly a half, and the " tick " will be materially 
quieted. 

The Gravity Escapement. 

This name is given to escapements where the fall of a weight 
through a definite height imparts the impulse, all impulses being 
thus equalized in intensity. A spring may be used instead of 
gravity to measure the definite impulse. Such are sometimes called 
isodynamic escapements. 

A gravity escapement employed by Wm. Bond & Sons in 
chronographs, and called isodynamic, is illustrated in Fig. 262. 
The same has been used in tower clocks with good results, and it is 
believed to be about the best in kind and in construction for that 
purpose, being the same in principle as the Bloxanr's or Dennison's. 

At A is the driving axis, on which is made fast a collar with an 
eccentric pin, a, and an arm, GH. A T-shaped gravity piece, DEF, 
is suspended by a spring, N, from a fixed clamp, QR, and will swing 
right and left, and it may rest against the eccentric pin a, or it may 
rest by its pin at F against the pendulum rod P. At D is a pin 
flattened on the lower side normal to a line DN. A second T- 
shaped gravity piece, IMK, is suspended on the other side and 
like the first except they are rights and lefts, and the pin at I is 
flattened on the upper side normal to a line IS. 

The pendulum rod P is suspended from the clamp QR by a 
spring, not shown, which allows it to swing to the right and left. 

The eccentric pin a throws the gravity pieces to the right or 
left. As shown, this pin a holds the gravity piece DEF in the 



230 



PRINCIPLES OF MECHANISM. 



extreme position the pin can give it; and the shaft A is locked in 
that position by the end of the slender arm GH striking against the 
detent pin at D. Now, as the pendulum rod, moving toward the 
left, strikes the pin at F, the gravity piece DEF is carried along 
with the pendulum to its limit of movement, thus releasing the 
arm GH, when, on making a half -turn, it is arrested by the end of 
the arm GH meeting the detent pin i, which now will be in the 



Q 



N 



E 



R 



"">. 



L 






M 



/ 



SO 






m 



isi 



^wJ 



-< 



O! 



■L 



Fig. 262. 
dotted position J", because the eccentric pin a has moved with 
H to the dotted position b and thrown the gravity piece IKM into 
the position JLM. This detent pin / detains the arm GH till the 
pendulum rod, on returning from its extreme position to the left, 
meets the pin L, and carries JLM along with it, thus releasing the 
arm H from the detent pin J, when the arm, the shaft A, and the 
eccentric pin a make another half-turn. 



ESCAPEMENTS. 231 

Now it is readily seen that as the pendulum rod returns, it is 
followed by JLM to the position IKM. That is, the pendulum rod 
takes the gravity piece from JLM to the limit of movement and 
back to IKM, one operation neutralizing the other as far back as to 
JLM-, but from the excess movement JLM to IKM the pendulum 
receives its impulse. A like impulse is received from F in the 
opposite movement of the pendulum. The eccentric pin a, moved 
by the train of wheelwork, raises the gravity piece from K to L, 
or lifts its center of gravity to create the impulse. 

The impulses imparted to the pendulum are thus made practi- 
cally equal for all time of running of the clock. 

The Cylinder Escapement. 

The cylinder escapement, formerly much used in Swiss watches, 
and interesting from the standpoint of mechanism, is illustrated in 




Fig. 263. 

Fig. 263. At A is the driving staff carrying the escape-wheel with 
peculiar shaped teeth having inclined edges, CD. These teeth act 
upon pallets at C and E, consisting of the smoothed edges of a 
thin half -cylinder, CFE, supported to swing upon a central axis, B. 
The diameter of the inside of this cylinder is just sufficient to allow > 
the latter to swing over the tooth CD with a trifle of clearance, ' 
and the distance CH, equal to 1G, should be just sufficient to admit 
the full cylinder with a trifle of clearance. 

The tooth CD is represented as escaping from the edge of the 
cylinder at C and soon to strike just inside the edge at E and to 
slide some distance, the cylinder turning right-handed. On 
reaching the limit of movement E returns, and on arriving at D 



232 



PRINCIPLES OF MECHANISM. 



the inclined edge DC of the tooth will slide against the edge E, 
imparting a left-handed impulse to the cylinder B. As the tooth 
CD escapes from E, the point H will strike just back of the edge 
of the cylinder at 67, and slide on the outside to the limit of move- 
ment, when on returning, as the edge G passes H, the inclined 
tooth 57" will slide along against the edge G and impart a right- 
handed impulse. On completing this, 
CD escapes at C as before. The 
amount of inclination of CD is arbi- 
trary, and it may be straight or some- 
what convex. 

There seems to be a large amount 
of friction in this escapement, parts 
being practically the whole time in 
rubbing contact, which circumstance 
may be sufficient to explain the un- 
spirited deportment of the escape- 
ment as observed in the watch. 
The Lever Escapement. 
This escapement, Fig. 264, is the 
one most used in watches and known 
also as the "anchor escapement," 
" detached escapement," or a detached 
lever escapement," because BH is 
a lever, or because DEBH resem- 
bles the anchor, or because the balance parts JKO are detached 
and free, respectively. 

This leaves the vibrating balance freest from friction of all the 
escapements, unless it be that for chronometers. As compared with 
Fig. 263, it has a very decided advantage in this respect, a fact, 
evinced by its more lively action as observed in watches. 

In this escapement A is the axis of the escape-wheel, D and E 
pallets of the anchor-piece DEBH swinging about an axis at B. 
The balance staff is at 0, upon which is a collar holding a pin at J, 
at which the collar is cut away nearly to the pin. The lever is 
slotted at I and has a pin or shoulder at H. At L and M are bank- 
ing pins or other provision against over-movement of the lever. In 
this there are really two movements, one the escapement proper 
between A and B, and the other a pin and slit motion between B 
and 0. Also a lock as between the pin H and the edge of the 
collar, to prevent BH from moving except at the proper time. 




Fig. 264. 



ESCAPEMENTS. 233 

Now supposing J to be moving toward / by the swing of the 
balance, when the pin /enters the slit, the pin H drops into the 
cutaway at J, and they will move along together toward K and be- 
come locked there upon the opposite side to that shown by the cut, 
and in a similar manner. When the limit of swing is reached J 
returns again to move HI back. Thus the balance staff and attached 
parts are free from contact with other pieces most of the time, since, 
when locked at the one side or the other, the escape-wheel teeth are 
locked upon the pallets as shown at F, where the angles are such 
as to draw D or E toward A during the lock and holding the pin 
H away from the edge of the collar J. 

When H moves to the right, the tooth at JV T is unlocked and its 
inclined end passes the end of the pallet at F, exerting a pressure 
upon it and imparting an impulse to the pin /. Similar action 
occurs at the pallet E to give an opposite impulse to J in the re- 
verse movement. 

The pallets, or the teeth, or both should be rounded by the 
amount of the angle of swing of B as shown, to prevent the corners 
of D or JVfrom scratching each other while in motion. 

In w r atches, D and E are jewel pallets and J is a roll jewel. 

The Duplex Escapement. 

This escapement is in use in expensive and also in cheap 




Fig. 265. 

watches, and is next to the detached lever and chronometer for 
freedom from friction. 

A, Fig. 265, is the escape-wheel, B the balance staff with a 



234 PRINCIPLES OF MECHANISM. 

longitudinal groove at H for a short distance to admit and pass the 
point of the spur H, J 9 etc., and D is a pallet to engage the impulse 
tooth F. 

The figure represents a spur just escaping the groove at H, and 
the tooth F about to engage with D, to impart to it an impulse 
while moving from D to the point of escape at E. Then the next 
spur, J, should meet the staff at I, leaving a trifle of clearance 
for E at G and F. The pallet moves on, to the limit of swing of 
balance and staff, and returns, passing to the other limit of swing, 
with J rubbing on the staff at / and skipping the groove H. Ke- 
turning from this limit, the spur J drops into the groove H, passes 
from I to H and escapes, etc., as before, the escape-wheel passing 
one tooth at each double vibration of the balance. 

The friction is chiefly the rubbing of the spur on the staff at 7, 
and to reduce this the staff should be made as small as consistent 
with its strength. 

This is properly called duplex, because of the two points of 
escape, the principal one at GE and the secondary one at H. 

The Chronometer Escapement. 

At A, Fig. 266, is the escape-wheel, B the balance staff, D the 
pallet, LGJKthe detent piece, G the detent, K the detent spring 




Fig. 266. 

supporting LGJ and forcing it against the banking pin N, IJ the 
feather spring bearing against the pin L in the detent piece, and I 
a projection on the staff B to act upon the feather spring. 

In action the projection / strikes the spring IJ resting against 
L, and forces LGJ away from the pin N, and the detent G away 
from under the tooth at G, when the tooth E strikes the pallet D, 



ESCAPEMENTS. 235 

imparting an impulse. As /? escapes at F, a tooth is caught by the 
detent G. The balance moves on to the limit of swing and returns 
with D just clearing F and E and with the projection /meeting the 
very light feather spring ILJ and flexing it to pass, when the 
spring flies back against the pin L. On reaching the opposite 
limit of swing, the balance returns, repeating the above movements 
in succession. 

To reduce the friction to a minimum, the staff and projection at 
/should be as small as admissible; also the feather spring and de- 
tent spring very delicate, especially the former. 

In watches and timepieces subject to abrupt displacement, the 
detent piece should be balanced, and perhaps pivoted, to prevent 
the detent from being jerked off its tooth at G. 

Also if balanced on a pivot with a retaining spring to hold it up 
to the pin N, it should be as light as possible to prevent abrupt 
turns in the plane of the escapement from unlocking the detent. 



PART III. 

BELT GEABING. 

Belt gearing includes all members in machinery concerned in 
transmitting motion in the manner of a belt and pulley, such as 
belts, bands or chains, pulleys or sprocket wheels for continuous 
motion; or for limited motion, where a rope, strap, or chain passes 
partly or several times around sectoral wheels, to which the ends 
are made fast, as in the windlass, or the "barrel and fusee" of 
chronometers, English watches, etc. 



CHAPTER XXI. 

VELOCITY-EATIO VARYING. 

THE GENERAL CASE. 

The Velocity-ratio. 

In Fig. 267 take the irregular rounded pieces shown as fitted to 
swing about axes at A and B, with a flexible connector DE passed 
over and beyond those points, and made fast to the rounded bodies, 
so as to admit of motion to some extent by A pulling DE and 
driving B. 

In a small displacement where D moves to G, E will move to J, 




Fig. 267. 

and regarding the flexible connector as iuextensible in length, JK 

236 



BELT GEARING. 23? 

will equal GH. Also the triangle HGD is similar to FAD, and 
KJE similar to IBE. 

If V and v be the angular velocities of A and B, we have 

V.AD = DG, v.BE=EJ, 

DG_AD EJ BE 

EG ~ AF' JK ~ BV 

, . ,. V DG BE AD X]n BI I BE 

and velocity-ratio =-=—-. —— = — - . HG • -^r^ . -^ —rj^ 

J v EJ AD AF EB JK AD 

BI _ BC 

~ AF~ AC 

which shows that the velocity-ratio equals the inverse ratio of the 
segments of the line of centers as formed by the intersection of the 
prolonged line of centers and connector. 

The line of the connector DE is here the line of action, and the 
velocity-ratio is the same as found in all previous cases, viz., equal 
the inverse ratio of the segments of the line of centers, counting 
from the centers of motion to the intersection with the line of 
action. 

The same is true if the connector cuts the line of centers be- 
tween A and B, in which case the directions of motion are contrary, 
while in the outside intersection they concur. 

For velocity-ratio constant the point C must be stationary, a 
condition readily secured for circular pulleys; also, it can be real- 
ized for a relatively long distance between centers for chain and 
sprocket wheels even with the latter non-circular, if symmetrical 
and with axes of symmetry at right angles, and for velocity-ratio 
equal to 1, 2, 4, etc. For the value 2, the larger wheel may be 
nearly a square, and the smaller nearly a duangle. But, as no ad- 
vantage for this combination over circular pulleys is conceivable, 
they will not be further studied here. 

But non-circular pulleys for velocity-ratio variable have been 
used with advantage. 

Law of Perpendiculars Given to Find the Pulley. 

In Fig. 268 take A for the axis of the non-circular pulley and 
the periphery e,f, g, li, to be found when the line of action Dg, or 
of the connector, has a known perpendicular distance, Ag, from A 
for each 1/8 turn of A. 



238 



PRINCIPLES OF MECHANISM. 



Lay off the several perpendiculars as radial distances, Aa, Ab, 
Ac, Ad, Ae, etc., and draw circles or otherwise transfer them to the 
several radii, marking the several angles through which A turns for 
the corresponding perpendiculars, giving points e, f, g, h, i. 
Through these several points draw the perpendicular line of the 
connector, as at e the line perpendicular to Ae, at /the perpendic- 
ular to Af, etc., to represent the position of the line of action for 
the respective radii. 

Then draw the curve of the pulley tangent to these lines of 




Fig. 268. 

action. The pulley outline does not pass through the several 
points h, g, f, etc., but puts the line of action at the proper per- 
pendicular distance from A. 

Some examples may serve to illustrate. 

VELOCITY-RATIO VARIABLE. 



DIRECTIONAL RELATION CONSTANT. 

1st. Example of the Equalizer of the Gas-meter Prover. 

The meter prover consists of a hollow cylinder of some 20 cu. 
ft. capacity, to be raised and lowered in water for shifting air through 
any gas meter for testing it. The more the cylinder shell is lowered 
into the water, the greater the buoyancy, this being counteracted 
by a weight suspended from the periphery of an eccentric pulley. 

To determine this pulley, let DE, Fig. 269, represent the descent 
of the cylinder shell into water, the surface of which is at Ee, equal 
steps of submergence being noted at points d, c, b, a. 



BELT-GEARING. 



239 



This cylinder is suspended from a circular pulley, AF, of such 
size as to make a 3/4 turn for the 
range DE, upon the axis of which 
is mounted the eccentric pulley, 
e, f, g, h, i, of a like 3/4 turn, to 
which a weight, W, is suspended. 

By trial or otherwise find the 
length of the perpendicular Ae, 
at the end of which the weight 
W will just balance DE in the 
highest position. Similarly find 
the perpendicular distance, Aa, at 
which W will just balance the 
submerged cylinder at the lowest 
position, or with D and a down to 
the water surface, Ee. 

Then divide ae = BE into the 
same number of equal parts by 
points b, c, d, etc., as there are equal 
angle divisions in the 3/4 turn e, g, i. Then make Af= Ad, Ag = 
Ac, Ah = Ah, etc., and through these points draw lines of action 
or perpendiculars to Af, Ag, etc. Now draw a curve as shown 
tangent to all these last-named perpendiculars for the linear profile 
of the eccentric pulley required. 

The parts being made and mounted as thus determined, there 
should be perfect balance between W and DE in any position. 




2d. Example of a Draw-bridge Equalizer. 

In Mehan's "Civil Engineering" is illustrated a draw-bridge, Fig. 
270, having a rope or chain with one end attached at a to the draw- 
bridge DE, while the other end winds upon a cylindric drum A, 
on the axis of which is an eccentric pulley, Fig. 271, upon which 
winds a rope or chain, to the lower end of which is attached a weight. 

Here the weight of the bridge is constant but causes a variable 
tension upon the rope Aa as the bridge is opened, which is to be 
equalized by the eccentric pulley and weight. Observing that the 
weight of the bridge moves in a circle a, b, c, about D, the rope 
tension for the points a, b, c, etc., will be 



tension = W 



Da 



W 



Dh 



Df> Dg> 



W 



Dh 



etc., 



240 



PKINCIPLES OF MECHANISM. 



which values are to be found and laid off on the line Aj, Fig. 271, 
giving points j, o, p, q, etc., for perpendiculars from A upon the 
line of action. Then draw an involute astu to the circle A, Fig. 
270, and take the distances bs, ct, du, etc., and lay off on the circle 




Fig. 270. 



Fig. 271. 



A of the drum, Fig. 271, giving angles jAk, j Al, jAm, etc., and 
draw the radial lines Aj, Ah, Al, Am, etc. Now lay off Jo on Ah, 
Ap on Al, etc., and draw perpendiculars to these radial lines 
through the points j, k, I, etc., and tangent to them the eccentric 
or spiral pulley required, as shown. 



3d. The Barrel and Fusee. 



In this the tension on the cord or chain, due to the torsional 
action of the spring in the barrel, is to be determined for equal 

angular positions of the barrel 
through the entire range of its 
motion. Starting with the spring 
slack or "run-down," find the ten- 
sion when just fairly started to 
wind up, as for the chain or cord 
Fig. 272. tangent to the barrel at D, Fig. 

272. Then pull E up to D and note the tension; then F to D, G 
to D, etc., noting the tension for each. These will increase as the 
barrel is thus wound up or forced around against the action of the 
spring. 




BELT GEARING. 241 

Through the center of the fusee at B draw an involute Ba to 
the barrel A. Then the distance from D to the involute Ba, fol- 
lowing the cord or chain, is always the same, whether the tangent 
point is at D or farther toward B. 

Then make the radial distances Ba, Bb, Be inversely as the 
tensions as above determined, Ba for tension at D, Bb for tension 
when E is at D, Be for F at D, etc.; and make the distances ab, be, 
cd, etc., equal the distances DE, EF, FG, etc. Through B and 
these points b, c, d, etc., copy the involute Ba, as shown, and at 
these points draw normals to the involutes. Then, tangent to these 
normals, draw in the curve of the fusee. 

In this construction, the smaller the angles DAE the more 
accurate will be the result until the limit is reached where the in- 
accuracies of graphical work predominate. For relatively large 
angles, DAE, etc., it will be advisable to make the distances ab, be, 
cd, etc., which are chords to the arcs, equal the straight dotted line 
or chord DE. Fig. 272 is made as if aD were to act by compres- 
sion. To change the figure for tension in aD, place b, c, d, etc., to 
the left of Ba. 

The " snail" used on spinning mules is like a double fusee, wind- 
ing a cord from small to large and then back to small radii again, 
thus varying the velocity of the cord taken up; or of the snail, when 
the cord is made fast at the remote end. 

In deep mines, where heavy wire ropes are used for hoisting, the 
winding drums may be made conoidal to compensate for the vary- 
ing load due to the varied weight of rope run out as the hoisting- 
cage is let down, and vice versa. If the length of rope run out per- 
revolution were constant, the drum would be a cone; but as more 
rope is let off per revolution where the drum is larger, its shape will 
be a concave conoid. 

4th. Non-Circular Pulley for the Rifling Machine. 

There was in use in the rifle factory of Windsor, Vt., fifty years 
ago, a rifling machine in which was employed a belt and non-cir- 
cular pulley connection, for the purpose of imparting to the rifle 
groove-cutting tool, held in a rod, an approximately uniform motion 
forward and back as the tool and rod traversed the rifle barrel. 
The rod was driven by a crank and pitman, on the crank shaft of 
which was an elliptic-shaped pulley similar to Fig. 275 connected 
by belt with a circular pulley above. For the slow motion here, 
employed this belt and pulley combination worked satisfactorily. 



242 



PRINCIPLES OF MECHANISM 



To determine the correct form of non-circular pulley on this 
crank shaft to give a uniform motion to the slide, consider the pit- 
man of infinite length, when the 
projections of the several positions 
of the crank pin a, I, c, g, h, i, etc., 
upon the line DE, Fig. 273, will 
divide the latter into equal parts, 
r, s, t, etc., representing the uni- 
form motion of the slide, as if the 
crauk pin F of the crank BF 
were moving through those points. 
Drawing Bg, Bh, etc., we obtain 
the angles the crank must pass 
Fig. 273. in equal time. On these lay off 

the respective perpendiculars br, cs, etc., giving Bn, Bm, Bl, etc. 
At the ends of these draw perpendiculars as in Fig. 268. These 
will all pass through the point a, as na, ma, la, etc., and drawing in 





Fig. 274. 

the pulley curve tangent to these perpendiculars gives simply the 
point a, which shows that the resulting non-circular pulley is merely 
a flat bar, FF, Fig. 274. Though this seems at first unreasonable, 
it is correct, since the belt from the uniformly-moving circular 
driving pulley A, going to the end F of the pulley FF, would move 

.F uniformly in the direction of FG, thus 
giving the crank pin at F the motion re- 
quired. Hence the pulley FF should be 
a flat bar with a very rapid motion when 
the crank pin is on the line of centers. 
This pulley would work with a suitable 
distance between centers without a serious 
variation of tension of belt. 
At the dead centers of the crank, the jerk of speed the theory 
would give may be avoided by arbitarily widening the pulley some- 
what as in Fig. 275. For moderate speed, as for a rifling machine, 




Fig. 275. 



BELT GEARING. 



243 



this belt and pulley combination would doubtless be thoroughly 
practical. 

DIRECTIONAL RELATION CHANGING OR CONSTANT. 

Prof. Willis has devised a movement answering to directional 
relation and velocity-ratio varying, called " cam-shaped pulley and 
tightener pulley," where any non-circular pulley may be mounted 
on any center and mated with a fixed circular pulley and a tightener 
pulley on a lever. The eccentric pulley should be the driver to 
avoid slipping of belt. (See Willis, page 201.) 



Example of a Treadle. 

One application of a wrapping connector with directional 
relation changing is found in the foot-lathe treadle, where an 
eccentric pulley, E, is put on the driving 
shaft, and a centrally mounted pulley, F, 
on the treadle bar, BD, and the two con- 
nected by a belt, the arrangement serving 
to avoid making a crank in the driving 
shaft, as in Fig. 276. The action is evi- 
dently the same as if a pitman were used 
to connect the centers E and F. 

It is not necessary that the pulley E be 
an eccentric, but it may be an ellipse, 
with A in one focus, or a flat bar clamped 
at one end to the shaft A. 

In case of the ellipse of the same length 
as the diameter of the eccentric, and with A at the same distance 
from the center, the full stroke of the treadle bar will be the same 
for both, but the ellipse will lower the treadle more slowly at first 
and more rapidly in the last part of the stroke. If A be at the 
center of the ellipse and the latter considerably longer than the 
diameter of F, the treadle will have two short double strokes per 
revolution instead of one. 




Fig. 276. 



CHAPTER XXII. 

DIRECTIONAL RELATION CONSTANT. 
VELOCITY-EATIO CONSTANT. 

Heke the pulleys are botli circular, of any relative dimensions, 
at any distance apart, and with belts open or crossed as in Fig. 277. 




Fig. 277: 

BC 

The velocity-ratio is always equal to —r-^ whether the belt is 

open or crossed, with C outside or between centers. When 
is outside, the pulleys turn the same way, and when between, they 
turn in opposite directions. 

The exact velocity-ratio is difficult to obtain, for two reasons: 
first, the thickness of the belt adds somewhat to the practical 
diameter of the pulley, and relatively more for the small than large 
one; and second, the elastic yielding of the belt, contracting as it 
goes around the driving pulley, and expanding as it goes around 
the driven pulley, causing a " slip " of belt which increases with 
the driving load and slackness of belt. This slip keeps the pulleys 
bright. 

The Belt. 

The belt may be a strap of leather, or of woven stuff sometimes 
filled with rubber; or of leather links on wire joint pins built up to 

244 



CIRCULAR BELT GEARING. 



245 



any desired width; or a strap with triangular blocks attached to run 
in a V-grooved pulley; or a round belt formed of thick leather, first 
cut into square strips and then rounded by drawing through a die 
when small, as for driving sewing machines ; or when larger, from 
f of an inch diameter up, it is made of a flat strip twisted and 
pressed; or a rope of hemp or wire. All but the flat belts run in V 
grooves. 

Chains running over sprocket wheels now constitute a most im- 
portant connector. 

Retaining the Belt on the Pulley. 

A V groove of sufficient depth will naturally retain the round 
belt or rope. But for the flat belts, the pulley is made " high cen- 
ter," since a flat belt has a tendency to climb to the highest part. 
This is due to the edgewise stiffness of the belt, giving it a tendency 
to climb a cone of moderate taper. 

Thus, in Fig. 278, take OAB as a cone cut at the lower element 
and developed by rolling out to DE till coincident with the tangent 




d^-->. 



Fig. 278. 

plane ODE. Placing a strip of belt, DE, upon it, we see that it runs 
off the developed cone surface at D and E, though at the middle 
point it is parallel to the base. 

Eedeveloping the portion of conic surface CD, and the belt 
with it, the latter is seen to run off the base of the cone, getting 
more and more inclined the farther it goes. 



246 



PRINCIPLES OP MECHANISM. 



Hence it appears that a flat belt running on a pulley with a 
high center will climb to the highest point. 

Advantage may be taken of this fact to carry 
a belt, not very wide, at quite an angle to the 
shafts, to avoid an obstacle, A, as in Fig. 279. It 
is here only necessary to make the pulleys quite 
convex and a little larger at one end. 

Grossed Belts. 



In a crossed belt the latter must have a twist 
of 180 degrees between pulleys to keep the same 
side of belt to pulley. Thus the edge elements 
of belt must be appreciably longer than the cen- . 
tral ones, by the ratio of the hypothenuse of a 
triangle to its base, when the latter is the length 
of the free part of belt, and the altitude of the Fig. 279. 
triangle the half -circumference of a circle whose diameter is the 
belt width. Hence it is not feasible to run a wide belt crossed for 
a comparatively short line of centers. 




Quarter-twist Belts. 

A quarter-twist belt is subject to severe strains, due to distortion, 
as well as the crossed belt, more for relatively large pulleys and 
less for small ones, as illustrated in Fig. 280. 





Fig. 280. 



Fig. 281. 



Without a guide pulley, the belt will immediately ran off the 
pulleys if the latter are turned backwards. 



CIRCULAR BELT GEARING. 247 

By arranging the pulleys as in Fig. 281, and adding the guide 
pulley to hold the guyed part of the belt over near to the straight 
portion, the pulleys may run either way without throwing the belt 
off and with less strain on it. The guide pulley, however, must be 
placed in an awkward position, its central transverse plane to coin- 
cide with the center lines of the belt above and below that pulley* 

Any Position of Pulleys. 

The driving and driven pulleys may be placed in any possible 
relative positions and connected by a single belt, if four or less 
guide pulleys be provided, four for the comparatively simple case 
where the pulleys are on parallel shafts and not in the same plane. 
For more awkward positions it will usually happen that tangents 
to the driving and driven pulleys can be made to intersect, at each 
point of which a single guide pulley may be located, with meridian, 
plane coincident with the plane of the proper tangents. 

Cone Pulleys. 

Pulleys in a series of steps, as employed on lathes and their 
countershafts, by which the speed of the lathe may be changed 
from one constant speed to a number of others by shifting the belt 
are called cone pulleys or stepped cone pulleys. Not only lathes 
but a large variety of other machines require these cones, so that 
their correct construction is important. 

A common practice is to make these steps equal on one of the 
cones, when they will require to be equal on the other for a crossed 
belt, but for an open belt not for uniform tightness of belt. 

A little inquiry will satisfy the seeker after truth that the cone 
diameters should be such as to place the steps of speed in geomet- 
rical progression, that is, the relation of any one speed to the next 
should be the same as that to the next ; or the ratio of any two ad- 
jacent speeds should be constant throughout, as has been correctly 
maintained by Professor John E. Sweet. 

For instance, in a " back-geared " lathe, the ratio of speed for 
the first two sizes should be the same as for the last two sizes of the 
cone, in order that in back gear the ratio of speed may be pre- 
served. Thus, for simple ratios, suppose the cones give the geo- 
metric series of speeds as 1, 2, 4, 8. Then the back gear should 
continue this series as 16, 32, 64, 128; the ratio of any two adjacent 
speeds being 2. That due to going into the back gear should still 



248 



PRINCIPLES OP MECHANISM. 



be the same. Otherwise, suppose that the series of speeds were as- 
sumed as 1, 2, 3,4; when the biick gear will give 5, 10, 15, 20, if 
the gearing is such as to give the first figure, 5. Then the ratios of 
adjacent speeds will give the series of ratios 

2, 1.5, 1.333, 1.25, 2, 1.5, 1.333, 

a quite irregular set of figures. Granting that they are correct up 
through the cone to back gear, the values of the ratios decreasing 
gradually from 2 to 1.25. Then, instead of continuing on a de- 
crease, there is a sudden jump to 2 again, after which a second de- 
cline — which is clearly irrational. 

It is also important that the gearing of the back gear be proper 
to this rational geometric series of Prof. Sweet as well as the cones, 
that the ratio of adjacent speeds throughout may be a constant. 

A convenient way of realizing in a drawing this geometric series 
of speeds in laying out a pair of cone pulleys is shown in Fig. 282, 
where a length AD, measured by some scale, represents the revolu- 




Fio. 282. 

tions per minute of the countershaft; Ba, Bb, Be, etc., by the same 
scale, the speeds of the lathe or other cone; BH, BI, BJ, etc., the 
radii of the countershaft cone; AP, AQ, AR, etc., the radii of the 
lathe cone; and AB the distance between the axes of the cones. 
This may be correctly drawn as follows: 

From B draw a line Bf at any convenient angle. Draw lines 
ah and lib, then bg, ge, ef, etc., parallel to them respectively, when 
the distances Ba, Bb, Be, etc., will be in geometrical progression 
and will represent the geometric series of speeds of the lathe cones, 
Ba being the slowest, and the others Bb, Be, etc., being made by 
trial to agree with the series required. 



CIRCULAR BELT GEARING. 249 

Draw lines Da, Db, Dc, etc., and extend them to intersect the 
line AB, also extended. From these points of intersection draw 
lines TSE, URI, VEP, etc., tangent to the circle GM, drawing the 
first line through S so that AS may measure the desired smallest 
radius of the lathe cone; or, if preferred, the line VEP may be 
drawn instead, making AP the desired largest radius. When the 
first line is drawn, strike in, tangent 'to it, the circle GM, from a 
center on a line midway between AP and BH. The radius of 
this circle is 

.„ AB 

OM — — nearly, , 

and is to be calculated. The remaining lines, URI, QJ, etc., are 
drawn tangent to this circle, giving the radii of the lathe cone AP, 
AQ, AR, etc., and radii of the countershaft cone BH, BI, BJ, 
etc.; and the cones may be drawn as shown. 

If a line as Db does not intersect AB on the drawing board, we 
must make Bb : BJ : : AD : A Q, which may be done on another 
diagram. 

The diagram shows that 

AD : Ba : : AP : BE', 

that is, the speed of the countershaft is to the slowest speed of the 
lathe cone as the largest radius of the lathe cone is to the smallest 
radius of the counter cone, as it should be to accord with the 
geometric series. 

In a diagram for a crossed belt the lines all meet in one point 
near GE, while for the usual case of open belts the positions of the 
lines must be " doctored " by being drawn tangent to the circle GM, 
which shifting of the lines, however, does not interfere with the 
velocity-ratio, since the ratio AS to BE will always be the same 
when the line is drawn through the point T. 

"To determine the radius OM, let AR and Ar, Fig. 283, stand 
for the corrections of the radii R and r to account for foreshorten- 
ing due to the inclination of the belt, which amount call Al. Then 
as TW is the belt inclination, half the foreshortening due to it is 



kW=XW-SE= 1/2 Al nearly, and = £ (AR + Ar), 



250 
whence 



PBIETCIPLES OF MECHANISM. 



# = *<!& + *} 




Also, by Fig. 283, 



Al R-r „ ._ . 

-g- : — H — ::-» — *•: ^ii> nearly, 



giving C#-r) 3 = 4l.AB = 2?rGF. AB. 

Again, Fig. 282, 

2GF:FM::B-r:AB nearly, 
whence 

^£.26^ 



B-r = 



FM 



Eliminating i? — r, we get 



But 
giving 



FM* _AB 
2GF ~ n ' 

0M:FM::FM:2GF nearly, 



0i/ = 



i?Jf ^5 



2£.# ?r 



the required radius. 

This radius, or rather the position of the point 0, Fig. 282, 
differs from that adopted by Mr. C. A. Smith, Trans. A. S. M. E. r 



CIRCULAR BELT GEARING. 251 

Vol. X., p. 269, where LO = 0.3UAB. The latter may be used 
in constructing Fig. 282, if preferred. 

In the extreme example of AB = 48", AS = 1", BH = 17.284", 
while an intermediate pair of sizes were 10", the difference of belt 
length, as given by Fig. 282, and the carefully calculated value, 
differed by. only about 0.2". 

Eeuleaux, in The Constructor, p. 189, gives an interesting dia- 
gram for determining cone pulleys; also Prof. J. F. Klein, in 
Machine Design, diagrams and tables; but in none of these do we 
find the very important consideration of the geometrical series of 
speeds. 

In practice the drawing for Fig. 282 may take an uncomfortable 
length. It may be shortened to the extent that the angle BTH 
does not much exceed 30 or 40 degrees, by taking for the actual 
distance AB a fictitious length ab, and using for the radius OM 
the value 

OM' = W 
nAB 

Actual cones have been employed where a variation of speed is 
desired while running, the belt being shifted when desired. It is 
usually difficult to keep the belt running satisfactorily, unless the 
cones are unduly long. 

TJie Evans friction cones is a good arrangement where cones are 
required, in which the cones are placed with large and small ends 
opposite, and a hoop of belting around one cone, somewhat larger 
than the large end, arranged with a guide, which hoop makes the 
bearing point between the cones. By shifting the guide the hoop 
is shifted and the speed changed. 

Rope Transmission, Rope Belting, Etc. 

i . The term rope transmission was first applied to rope belting 
used for transmitting power over relatively long distances, but re- 
cently it has come to be employed as a substitute for leather belting. 
For transmission wire rope has largely been employed without a 
" take-up " where the sag of the rope for stretches of several hun- 
dred feet will vary, compensating for temperature and wear, and 
still maintain sufficient tension for service. With this tension and 
a speed of the rope varying from 3000 to 5000 ft. a minute, a large 



252 PRINCIPLES OP MECHANISM. 

amount of power will be transmitted with only a half -turn of rope 
over the pulley for frictional contact. 

But for shorter spans, where hemp or cotton rope is used, provi- 
sion is made, first, for " take-up " for maintaining constant, or at 
least sufficient, tension; and second, for drawing the rope as it 
stretches. In cable railways, where the working tension of the rope 
is carried to a very high value with comparatively low speeds, the 
take-up and increased frictional contact with the driving drum are 
matters of the utmost practical importance. In the slower speed 
arrangements the rope is to be consumed by a higher working ten- 
sion, while in the higher speeds the rope is worn out by a high 
working velocity, causing frequent flexural stresses and abraiding 
contacts at the pulleys. 

In Rope Transmissions there are two leading systems : first, where 
the one loop of rope reaches the entire stretch of the system, with 
a driving pulley at one end and a driven one at the other, guide 
pulle}"s of less diameter being introduced at points between to carry 
the rope, the whole layout of rope being in one vertical plane; and 
second, where one loop of rope passes over a driving and driven 
pulley only, with no guide pulleys, beyond which is a second loop 
of rope, likewise mounted and driven from the first, and beyond 
which last is a third loop of rope similarly mounted and driven from 
the second, and so on for as many bents as desired; there being at 
each intermediate station point either two pulleys made fast on one 
shaft or a double pulley. This system must be arranged in one ver- 
tical plane, but, as in the first, may pass over hills and valleys. 

The driving and driven pulleys are of large size; first, because of 
the high speed required, and second, to prevent a too severe flexing 
of the ropes. The bottoms of the grooves in the pulleys are fitted 
with wood, hard rubber, leather, or like material, to increase the. 
friction of contact with the rope. 

A horizontal angle may be turned in either of these systems in 
at least two ways: first, by introducing bevel gears at any station; 
second, by passing a rope vertically downward or upward from one 
pulley and immediately upon another set in a vertical plane making 
any desired angle of deflection with the first plane. 

In the above, wire rope is usually employed. 

In Rope Belting several half-turns are made around the driving 
and driven pulleys to produce the necessary frictional contact. This 
is done differently for long than for short stretrhes between power 
pulleys. 



ROPE TRANSMISSION. 



253 



First. For a Short Stretch, Fig. 284 illustrates a mode of dupli- 
cating the passes of rope from driving to driven pulleys. The rope 
passes from the take-up or tension pulley D to the first groove of 




B. Passing half around B it runs to the first groove of A. Passing- 
half around A it goes over to the second groove of B and a half 
around, when it goes to the second groove of A, and so on for all 
the grooves of A and B till when it leaves the last groove of B it 
passes to and half around the take-up pulley D, when it goes to the 
first groove of B again and repeats the circuit. One piece of rope 
spliced into a single loop makes the entire circuit. 

The take-up pulley D is mounted upon a slide, and has a weight 
to produce the desired tension and take up the slack due to varying 
length of rope. 

If the pulley grooves of A are all of one diameter, likewise of 
B though it may differ from that of A, the ropes in the working 
tension side will all have the same tension, and those on the slack 
tension side will be in equal tension with each other, but less than 
that on the working tension side, but not if the grooves differ in 
diameter on either one of the wheels. 

The pulley D may be placed outside of A and B, as if the rope 
passed from B over beyond A and then back to B again, but with 
no advantage. 

Second. For a Longer Stretch, the rope belt may pass over a 
counter pulley at each end of the system, as in Fig. 285. 

Here the rope between A and B is in higher tension than in 
Fig. 284, it being due to the cumulative action of all the half-turns 
of rope contact with the driving pulley. 

One point to be noticed here is the fact that the rope is varying: 



254 



PRINCIPLES OF MECHANISM. 



its tension, or seeks to, from the first groove of A to the last, and 
consequently varying its length. On this account the pulley A 
should have its grooves varied in diameter from one end to the 
other, and likewise for the counter pulley D. The same should be 
done at the end B. The counter pulleys D and E, however, may 
consist of separated sheaves loose upon the axis and of one size. 
The amount the grooves of A and B are to vary in diameter will 
depend upon the unit of elastic yielding of the belt, and the total 




Tariation of tension between the going and returning sides. The 
drop in tension from one end to the other of A or of B should be 
uniform per groove, and likewise the diameter; while the end 
diameters should differ by the amount an equal length of belt will 
vary as due to the total variation of tension. 

The counter pulleys D and E may both be stationary, with a 
single turn of belt going over to a single-groove tension pulley F t 
Also, D and E may be geared up with A or B, and become power 
pulleys. 

Third. For Cable Railways, Haulage Lines, etc., the pulley and 
counter pulley A and D are connected and driven by power, while 
at B there is only a single sheave for returning the rope. 

Ingenious compensating devices are in use to provide for the 
variation in rope length while passing over the series of grooves on 
the connected driving and counter pulleys. 

For instance, let G and H, Fig. 286, represent two pulleys of 
equal size and number of pulley grooves, tapered from H toward 
G to partially compensate for varied rope length between H and 



CHAIN AND SPROCKET. 



255 




Fig. 286. 



G. The two pulleys are loose on the shaft F, but driven by a 
bevel gear at IL and JK, mounted 
upon axles at right angles to the 
shaft F, and made fast upon it. 
These gears mesh into a bevel gear 
IJ fast in the pulley H at IJ, and 
into a bevel gear LK fast in the 
pulley G at LK Thus mounted, 
if H is turned one way, G will be 
rotated in the opposite direction, by 
reason of the bevel gearing inter- 
posed. 

A second system like Fig. 286 may be prepared and the two 
connected by gearing to serve as A and D in Fig. 285. When 
rigged with rope and started into service the varied diameter of H 
will partly compensate for varied tension and length across H, and 
likewise across G ; but between H and G the gearing will permit a 
perfect compensation, by revolving slowly in opposite directions. 
Then, when the rope is working under a greater tension than the 
taper of pulleys H and G provides for, they will move one way on 
the shaft F, and the opposite way for a less tension than the taper 
of pulleys provides for. Thus the creep of rope at contact with 
pulleys is reduced by the splitting of the pulleys from one into 
the two, G and H, 



The Chain and Sprocket Wheel. 

A wrapping connector, connecting wheels with an exact velocity- 
ratio, is found in the chain so formed that it will engage the teeth 
of a spur or sprocket wheel, as in the familiar example found on 
the safety bicycle, for a connection between pedal axis and driving 
wheel. 

The chain is made with various forms of link, one of the earlier 
ones being shown in Fig. 287, where the links are punched out of 
sheet metal and pinned by rivets, while a more modern one is 
shown in Fig. 288, all links being of the same form, sometimes 
cast and sometimes drop-forged. The latter possess important 
advantages in having the axial pin solid with the rest of the link, 
and in having each piece of the chain like every other, all hooked 
together. A link, turned into the position shown at D, may be 
removed from the hook and more or less links added, while for 
working positions the shoulders a and b prevent unhooking. It is 



256 



PRINCIPLES OF MECHANISM. 



thus easy to shorten a chain when worn and slack, and by the 
amount of one link length; while for Fig. 287 a smith is required^ 
and the least that can be removed or added is two link lengths. 





r i h i i i i i i ■ 1 1 i j i ■ m 



Fig. 288. 



The Teeth of a Sprocket Wheel engage at E, F, etc., or H, I, J T 
etc., Figs. 288 and 289, as the case may be; the proper forms of 




Fig. 289. 



the teeth being determined as shown in Fig. 289. 

Take HGF a portion of the chain on the wheel, while at F it 
runs off in the tangent FED. 

Now as FD is wound upon the wheel, the center point E de- 
scribes a circle arc about F till the link head at E strikes the rim 
of the wheel, when the links EDI will swing about li as a center 
till D strikes the rim of the wheel, and so on. Then we will find 
that the arc from a to the straight line FJ is circular about h as a 



CHAIN AND SPROCKET. 257 

center; from F 'to GFK it is a circle arc with F as sl center; and 
from K to L it is a circle arc about G as a center, etc. Then a 
series of circle arcs ef,fg, parallel to ab, bd and at a distance ^ 
from it, will be proper for a sprocket tooth profile for the circular 
wheel A. This curve copied around will give all the teeth. 

For non-circular sprocket wheels, as in the cases of the elliptic 
ones used on some bicycles, the sprocket-tooth outlines, to follow 
theory exactly, should each be determined in the manner of Fig. 
289. 

Non-circular sprocket wheels have a variation of velocity-ratio 
the law of which is generally simplest when one of the pair is cir 
cular; but both may be non-circular and in any ratio of sizes, pro- 
vided the wheels both have symmetrical axes at right angles to each 
other, observing that the pair of wheels should be so far apart that 
the inclination of the chain to the line of centers does not become 
so excessive as to vary the tension of the chain unduly. 

Practical Application of Chains. 

Chains are often used to retain a mathematical relation between 
a pair of axes where other wrapping connectors would not answer. 

They have been tried for the transmission of power, but experi- 
ence shows that they must run so very slowly to prevent noise and 
shocks that they have been abandoned for this purpose. 

Hooks and projections have been formed, on the links to which 
may be attached conveyor bars, boards, buckets, scoops, etc., when, 
in stretches of considerable length, sometimes several hundred feet, 
running vertically or on inclined tracks, grain, sand bags, refuse, 
etc., may be carried in a continuous current. Thus excavators 
have been operated, and coal-cutting machines, where cutting tools 
are made fast to the links. 



PART IV. 
LINK- WORK. 

This term is applied to such machinery as consists of rods, 
cranks, levers, bars, etc., jointed together, either for axes parallel 
or meeting, or crossing and not meeting. 



CHAPTEE XXIII. 

THE GENERAL CASE. 

The Velocity-Ratio. 

Take A and B as fixed centers of motion, AD the driving crank 
or lever, BE the driven crank, and DE the connecting-rod bar or 
link. As AD turns about its center A, the rod DE compels BE to 
turn also ; Fig. 290. 

To determine the velocity-ratio compare with Fig. 267, in which 




Fig. 290. 

AD, DE, and BE may replace the lines of like lettering of Fig. 290. 
Hence for the latter the 

V BG 

velocitv-ratio = — = -77-. 
v AC 

258 



THE GEXERAL CASE. 259 

When the point C is beyond AB the cranks turn in the same di- 
rection, and in opposite directions when it is between. 



Peculiar Features of Link-Work Mechanism. 

Link-work movements are the most arbitrary in law of motion 
of all the combinations of mechanism, and in many cases where it 
would otherwise be preferred, must be abandoned on this account. 
It is the "lightest-running" mechanism known, the resistance 
being due to the slight friction of comparatively small pins in bear- 
ing holes well lubricated, and with comparatively long arms with 
which to turn those pins. Also, a pin rarely makes more than one 
complete turn in its bearing hole in a complete movement, and 
usually much less; while in the corresponding movement of a cam 
motion the roller (as in the best arrangement) makes from 6 to 12 
turns on its pin, and even this is not so prejudicial as regards re- 
sistance as the rolling of the roller along the surface of the cam 
groove. 

Link-work is consequently much more durable than other forms 
of mechanism, and should be adopted into machinery whenever it 
can be in preference to cam-work, tooth-gearing, etc. 

It often happens that the principal or hardest working move- 
ments of a machine may be link-work while cam motions adapted 
to it may serve for other movements, and give, on the whole, a 
more satisfactory machine in operation than where all are cam 
movements. 

Here, instead of assuming a law of motion and finding the link- 
work, as can be done in gearing, cam and belt movements, etc., we 
must examine a proposed link-work movement complete, when, if 
found unsuitable, try another, etc., till an acceptable one, if possi- 
ble, is obtained. 

The motion sought may be impossible, while an approximation 
to it may answer by conceding some unimportant idiosyncrasy of 
movement. 

I. Axes Parallel. 

Most link-work belongs to this case of axes parallel, and the 
treatment must largely be by examples, owing to peculiarities and 
want of susceptibility of reduction to broad and extended laws. 



260 



PRINCIPLES OF MECHANISM. 




Some Examples of Link-work Movements. 

Example 1. — To illustrate: An excellent needle-bar motion 
for a shuttle sewing-machine is obtained by link-work; and it is, 
without question, in one case at least, the light- 
est-running shuttle machine yet built, the whole 
machine being link-work mechanism. 

The needle-bar motion referred to is shown 
in Fig. 291, used in several different sewing-ma- 
chines. At A is a rocking shaft with a crank 
arm AD attached, the latter reciprocating be- 
tween the angular positions AE and AF, the 
connecting link DB raising and lowering the 
needle-bar BJ. 

As D passes the line A I of crank and link 
straightened, the bar reaches its lowest position. 
As D moves on to its limit of motion at F the 
bar is raised an amount GH, forming the loop 
for the shuttle. Now it is not necessary for the 
needle-bar to return to G, but on tarrying a 
moment raised to H for the shuttle point to 
fairly enter the loop, it may reasonably enough continue on its up- 
ward journey. By a cam it would be given this action, but by the 
link-work of Fig. 291 it must return to the lowest point again be- 
fore making its ascent. 

Now if this drop by the amount HG, after the shuttle enters 
its loop, is considered fatal to the machine, the proposed link move- 
ment of Fig. 291 must be abandoned; otherwise it may be accepted, 
giving a much lighter-running and more durable combination than 
if the cam and pin be adopted, or such a movement as in Fig. 252. 
Example 2. — Another instance is found in the Corliss valve 
gear, where the rocker plate swi 
forth from H to F. The valve and 
stem B are moved by the lever BE 
as it swings from BE to BG. As D 
moves from J to F and return the 
valve is opened and closed, and from 
J to H and back it remains closed, 
but is compelled to be moving 
slightly, as due to the double versed-sine DK. If this movement 
DEis counted out of the question, the link-work combination of Fig. 



to carry the pin D back and 




Fig. 292. 



EXAMPLES OF LINK-WORK. 



261 




292 must be rejected, otherwise accepted, as in fact it has been by 
hundreds of builders following in the footsteps of Geo. H. Corliss. 

Example 3.— Another example is given in Fig. 293, where the 
link-work movement rotates a shaft 
B, by the arm BE, from the position 
BF to BE and back, during the half- 
turn of the main shaft that drives AD 
from A I to AD and back; while for H 
the remaining half-turn of the main 
shaft, D moves from I to H and back, 
during which BF is nearly station- 
ary, its simultaneous movement being 
from F to G and back. It was pre- 
ferred that for this latter BF remain 
absolutely stationary, but the slight 
movement FG was counted less ob- Fig. 293. 

jectionable than cam-work, as compared with the linkage of Fig. 
293, so that the latter was adopted. 

It may be noted that the slight movement FG is less when the 
two arcs HI are convex the same way than when convex opposite 
ways, and more so as A is nearer to B. 

Example 4. — As another example, suppose a point is required 
to move from E to F, Fig. 294, and return within the sixth part 

of a turn of the main shaft A. A 
cam may be used, by which the point 
is thus driven and then allowed to re- 
main quiet at E for the 5/6 remaining 
turn. But if there is no particular 
objection to the movement of the 
point from E to G and return, instead 
of remaining stationary at E, link- 
work may be employed, as in Fig. 294, 
where A is the main shaft, AD the 
crank, De the pitman, and eBG = 
HBF a bell-crank lever, the arrange- 
ment being such that while the crank pin moves from c to b the re- 
quired sixth part of a turn of A, a moves to Hand back, and Eto F 
and back, thus meeting the essential conditions of the movement. 

Path and Velocity of Various Points. 

In the study of the motion of linkages it is sometimes necessary 
to determine simultaneous positions throughout the movement for 




Fig. 294. 



262 PRINCIPLES OF MECHANISM. 

all the joints, beginning with the driver which, for uniform motion, 
should have the points equidistant, as illustrated in Fig. 295. Here 
the driver A makes its circuit with uniform motion from 0, 1, 2, 
etc., to 8. Corresponding points are given like numbers throughout. 
The link BE swings about the fixed point B, and the end F oi the 




Fig. 295. 

link DEF describes the curve F, 5, 6, etc., and returns to F, all 
the spaces being passed in equal times provided A moves over its 
numbered spaces in equal times. 

The curve described by any point of the link DEF may be 
determined, as was done for the point F in the above instance. 
Modifications of the path F, 5, 6, etc., may be made by changing 
the position of B, or length BE, or angle DEF, etc. 

A link may be connected at F, thus extending the linkage into 
a train, and possibly several links in succession, leading through 
several elementary combinations. 

Sliding Blocks and Links. 

Sliding parts are sometimes introduced, either for realizing the 
conditions of a link of infinite length, or for simplifying the 
mechanism, a notable example being that of the crosshead of the 
steam-engine. 

Sometimes a link may be greatly shortened and simplified in 
construction, as shown in Fig. 29G, where the pin Amoves in some 
curve FE, as that described by the point F in Fig. 295. In fact 
the parts shown in Fig. 296 may be employed in continuation of 
those of Fig. 295 into a train, F moving through the points 1, 2, 3, 4, 
etc., in equal times, being regarded as the driver for the elementary 
combination of Fig. 296. The point G is the fixed fulcrum pin for 
the bell crank HGD, and points 1, 2, 3, etc., denoting positions for 
H, may be found as corresponding with like figures of the curve 
FE,sls if that curve were the one through i^of Fig. 295, or again, 
as corresponding with the points 1, 2, 3, etc., of AD, Fig. 295. 



EXAMPLES IN" LINK-WORK. 



263 



The block FD serves as a short link connecting the straight 
pins F and D, upon both of which the block slides to accommodate 
the versed sines of the curve FE and of the circle arc about G. 

As these pins are always perpendicular to each other, the block 
FD is a simple single piece with the two holes at right angles to 




Fig. 296. 

each other. In this way the motion of DH is to be found as in 
sliding contact, and on this account the movement would seem to 
require classification there. To determine the velocity-ratio, find 
the center of curvature B, of the curve FE at F, and draw the nor- 

jnri 

mal FC to the pin D, and the velocity-ratio = y^,. 
simple when the curve FE is a circle. 



This becomes 



CHAPTER XXIV. 

THE ROLLING CURVE OR NON-CIRCULAR WHEEL EQUIVALENT 
FOR LINK-WORK.— GABS AND PINS. 



Foe every elementary combination in link-work the equivalent 
motion can be obtained by wheels in rolling contact. 




Fig. 297. 
Example 1. — In Fig. 297 the axes A and B are connected with 
link work as shown, A G being the driving crank, MG the connect- 

264 



THE ROLLING CURVE EQUIVALENT FOR LINK-WORK. 265 

ing rod, and BM the driven crank. Their lengths are such that 
they will all come to coincide with the line of centers DA CEB. 

To find a rolling curve possessing the same law of velocity-ratio 
as the link- work, the links are to be put in several positions, as 
AG MB, etc., and the intersections with the line of centers found 
as at C for the position GM. Now the velocity-ratio in link- work 
being the inverse ratio of the segments AC and BC, also the same 
for rolling wheels in contact at C, it follows that a pair of wheels 
in rolling contact at C have the same velocity-ratio as the links 
AG MB, with intersection of link at C. Therefore, revolving C to 
T in BM prolonged, and C to P in. AG prolonged toward P, we 
have points T and P in the peripheries of the rolling wheels. Other 
points, Q, D, A and R, S, are found in the same way, when with 
points sufficient the curves may be drawn in complete, above and 
below the line of centers, giving the rolling-wheel equivalents of 
the link-work. 

The positions of these curves in the figure are proper to the 
crank positions of AD and BE, as if the wheels were finished and 




Fig. 298. 

made fast to the cranks in these positions. The distance BS = 
BA, where the rolling suddenly stops, corresponding with the limit 
of the crank movement where the crank AD and link take posi- 
tions on the straight line AO, BO being the extreme position of the 
crank BE when AG + GM = AO. 

In Fig. 298 is shown as separated from Fig. 297 the link-work 
and the equivalent wheels due to the intersections of the link DE, 



266 



PRINCIPLES OF MECHANISM. 



with the line of centers. Throughout the movement, the curves 
will remain continually tangent to each other at a point C, so that 
the point of tangency, and the intersection of DE with AB will 
remain some common point C on the line of centers. 

In Fig. 298 the intersection C is for the line DE with the line 
of centers AB. If BE be fixed for the line of centers, the inter • 




Fig. 300. 

section will be that of the lines AD and BE, both prolonged, the 
curves for which are LKN, UJV, Fig. 299; and Wa, Xb, and YBZ, 
Fig. 300, as given in Fig. 297 and shown separately in Figs. 299 



THE ROLLING CURVE EQUIVALENT FOR LINK-WORK. 



26 r 



and 300, all extending to infinity, due to FH and F'H' becoming 
parallel to BE. One position of the links is shown at BFHE , Fig. 
299, and another at BF'H'E', Fig. 300, the first giving an inter- 
section at I and the second at F. The intersection I is revolved to 
the positions BF and EH, Fig. 297, for points in the curve, while 
the point T is revolved to BF' and EH' extended backwards, be- 
cause this part of the curve is at the other side of the infinitely 
distant points. Thus Fig. 297 embraces the three sets of wheels 
of Figs. 298, 299, and 300. 

Example 2. — The rolling- wheel equivalent of the crank and pit- 
man is shown in Fig. 301. 

The crosshead A as driver has its center of motion at an infinite 
distance, so that the line of centers is BH extended to infinity. 

Taking the crank in the position B G and the pitman at A G, 
the intersection of the latter with the line of centers is at D in A G 




A J 



Fig. 301. 




Fig 302. 



prolonged. Eevolving D to the crank gives F for one point in the 
wheel B. Also revolving D to the line EA, representing the crank 
line from infinity, gives the horizontal line BE and the point E in 
the mating wheel A. Proceeding thus, we obtain the curves 
BFHB and JEHI for the rolling-wheel equivalent for the crank 
and pitman motion for a half -turn of the crank. The proper 
position of the crank relative to the wheel BFHB is the line BH. 
The curve IHJ is supposed to be made fast to the crosshead and 
moving with it, as shown in Fig. 302. 



268 



PRINCIPLES OF MECHANISM. 



Example Z. — In Fig. 303 we have an example of two cranks 
connected by a connecting rod and drag link, the connecting rod 




Fig. 303. 

having a fulcrum pin near its center carried on a swinging link, 
the lower end of which works on a fixed pin. 

For this somewhat complicated elementary linkage the rolling 
wheels are worked out and teeth set upon them for a pair of gear 
wheels, the same being combined with the linkage in the one 
movement for a practical illustration of the equivalence of link- 
work with its proper rolling wheels. 

The gears show that the velocity-ratio of the linkage is far from 
constant, though, judged from the link movement alone, might be 
taken to be nearly so. By measured radii, taking the driver to be 
moving at a constant rate, the ratio of the fastest to the slowest for 
the driven wheel is over 3. 

Example 4. — An interesting linkage is found in the Peaucelliers 




Fig. 304. 

parallel motion, called by Prof. Sylvester the 
Fig. 304 in the links ABDE. 



kite," shown in 



THE ROLLING CURVE EQUIVALENT FOli LINK-WORK. 269 

Making AB the fixed line of centers, and A the driver, E 
describes the circle EB about A, and D describes the circle through 
DG about B. 

Dotted positions of the links are shown to indicate the nature of 
the motion. The crank AE, it is readily seen, makes two revo- 
lutions to one of BE, and there are two positions for dead points, 
viz., when the joint E is at G, and at 180° from G. 

The equivalent rolling wheels are readily found, one point for 
each wheel being at the intersection J of EE prolonged, / being its 
correct position for A, with reference to the crank position AE, and 
also the correct position for B relative to the crank position BE- 
The curves being symmetrical, we may revolve J about A to /and J 
about B to A" in AE and BD prolonged, giving points / and iT in 
the wheels relative to the cranks for the positions AB and BG, as 
if the wheels were cut in material and made fast to the cranks, the 
smaller one to crank AB, and the larger to the crank BG. 

Thus determining a sufficiency of points and locating them all 
with respect to some one position of the cranks, we obtain the out- 
line of the wheels shown. 

The wheels fully worked out are shown together with the link- 
work in Fig. 305, where the B wheel is seen to make one complete 
circuit of the rim with a side offset in 
it suitable for the mating wheel, and 
where the A wheel makes two com- 
plete convolutions, one being out of 
the plane of the other to prevent in- 
terference and mating respectively 
with the offset parts of B. 

The wheels in this construction 
work by rolling contact of pitch lines, 
except at the dead points, where half- 
teeth are introduced as shown. 

Example 5. — The case of two equal 
cranks revolving in opposite directions FlG - 305 

and a connecting link of the same length as the line of centers, but 
longer than the crank, is shown in Fig. 133, accompanied with the 
equivalent wheels, the latter being the rolling ellipses. Hence the 
law of velocity-ratio for this linkage is the same as that of a pair of 
rolling ellipses, each with the axis at a focus. 

Example 6. — The same linkage as in Example 5, except that a 
shorter instead of a longer member is fixed to serve as a line of cen- 




270 PRINCIPLES OF MECHANISM. 

ters, is shown in Fig. 306, where the equivalent rolling curves are 
hyperbolas instead of ellipses. The hyperbolic wheel ECG is fixed 
upon the arm BE, while FCH is fixed on the arm AD. 

For the positions A /and BJ of the arms these hyperbolas are 
tangent at the point K of intersection of 1J with the line of centers 




AB, both prolonged. The points H and G in the hyperbolas are 
obtained by revolving iTto the radii ^4/ and BG. These hyperbolas 
will roll in mutual contact from where the link IJ is parallel to the 
line of centers, through 180 degrees, to where it is again parallel to 
the line of centers, when another pair of hyperbolas, shown dotted, 
will come into rolling contact for the remaining 180 degrees. 

The dotted hyperbolic half -wheel LM is to be fixed upon the 
arm AD, as well as the half -wheel FH, while the remaining two 
half -wheels, EG and NP, are to be fixed upon the arm BE, and at 
a distance between vertices equal AB. 

The points B and D are foci to the hyperbolas EG and FH. 
Also A and E are focal points for the dotted hyperbolas. Thus A 
and D are focal points for the hyperbolas FH and LM, the asymp- 
totes for which intersect each other, and the line AD at its middle 
point. 

A pair of these hyperbolic wheels mounted for non-circular 
wheel pitch lines is shown in Fig. 49, where AB, as in Fig. 306, is 
the constant difference of lines drawn from all points C to the focal 
points. 

The elliptic wheel of Example 5 and Fig. 133 may coexist along 
with the hyperbolic wheels of Fig. 306, except that one of the 
ellipses would be fixed with A and B as focal points, while the 



THE ROLLING CURVE EQUIVALENT FOR LINK- WORK. 271 

other would be carried on the link I J, their point of tangency 
being constantly the intersecting point of A I with BJ, as shown in 
Pig. 307, while the hyperbolas are tangent where AB and JI inter- 
sect at D. 

A most interesting and instructive figure in this connection is 
brought out by aid of his theory of Centroids by Beuleaux, and 
given in his Kinematics of Machinery by Kennedy, p. 194, much as 
shown in Fig. 307, where the linkage, the rolling ellipses, and hy- 
perbolas are all presented in one view. 

Thus with AB the line of centers, IJ may be a link connecting 




^ 



the focus J" of PiVwith the focus / of HD, as has been admirably 
shown by Geo. B. Grant in his handbook on Teeth of Gears, Second 
Edition. 

Examining Figs. 133 and 307, we find that for the ellipses both 
the link and line of centers in length equal the sum of the dis- 
tances from the point of tangency of the rolling ellipses to the pair 
of foci A and B on the opposite sides of the center of the ellipse. 
In the hyperbolas the link or line of centers equals the difference of 
these distances from the point of tangency. 

Referring these curves to their conic sections, we find that the 
parabola lies intermediate, and is like the ellipse of infinite length, 
or the hyperbola of infinite distance between foci. 

■Example 7. — Therefore we may expect that the link connecting 
foci of the parabolas would be of infinite length, as shown by Geo. 
B. Grant in his Teeth of Gears, Second Edition, and may be real- 
ized in the manner of Fig. 308, where AB is the fixed line of cen- 
ters, C the point of contact, A and D foci, DE a link sliding on a 
straight guide, mounted on the parabola A, perpendicular to its 



272 



PRINCIPLES OF MECHANISM. 



geometric axis and giving to D the same motion as if DE extended 
to infinity and were pivoted there to the axis or opposite focus of 
A, The parabola D slides on a guide FG perpendicular to the 




geometric axis of D. The point of contact C is at the intersecting 
point of the link and line of centers. 

In Fig. 309 we realize this link-work movement without the 
parabolas. The gabs and pins will be explained later. 

Example 8. — The above examples are all for curve equivalents 




Fig. 309. 
that are symmetrical; Example 2, apparently not, becomes so when 
the remaining half-revolution is provided for. 



THE ROLLING CURVE EQUIVALENT FOR LINK-WORK. 



273 



As an example entirely wanting in symmetry, take the link-work 
of Fig. 293, the rolling-curve equivalents for which are shown in 
Fig. 310. 

The links are shown in the position such as places the joint E 




Fig. 310. 

on the line of centers AB, and in the proper relation to the curves 
for mounting them upon the crank arms, IL and OP on AE, and 
JK and MN on BD. 

To find a pair of points in the curves JE and IL: Place AE in 
the position Ab, when the link ED will follow to the position bc y 
giving the intersection G with the line of centers. Then, as before, 
revolve the point G to the arm Ab, giving the point e. Now if 
the curves were symmetrical this point could be retained as a point 
in proper location of the curve. But here e must be placed at/ 
symmetrically on the opposite side of the line of centers, so that 
when the arm moves from E to b this point will move up to the 
line of centers. Also, the mating point a, is to be placed at i, a 
being on a line Ba, where the angle CBa = DBc. In like manner 
find other points. 

These curves reach infinity when the link DE becomes parallel 
to the line of centers, following which the curves MN and OP 
come in from infinity. The point of OP at A is in contact with 
its mate when the link DE is in a line running through A; the 
portion A answering to the slight return of D from its extreme 
point, as E moves from the straight line DA toward the extreme 
point F. For the other extreme, G, of movement of E, the points 
/and /are in contact on the line of centers. The movement of 
D is slight for the movement of E from R to F, while for RG 
the crank B makes over a 90-degree movement. The point R is 
about midway between F and G, so that D is comparatively quiet 
for half the time. 



274 



PRINCIPLES OF MECHANISM. 



Dead Points in Link-Work. 

A dead point or dead center, in link-work mechanism, is a point, 
or set of points, or positions of the links, at which, if certain of the 
links in combination be made driver, the linkage will be found 
positively locked. Thus when the crank and pitman are in line 
the crank cannot be started into motion by force applied to the 
crosshead. Dead points must always be provided against, either by 
inertia, by springs, or by extraneous attachments. 

In the steam engine the inertia of the fly-wheel serves. In 
some reverse motions in machines springs are employed. In start- 
ing inertia is dead also; and so the single-acting steam engine must 
not stop on the dead center. In locomotives, two sets of cranks 




Fig. 311. 

are placed at right angles, so that one is at its best advantage when 
the other is on its dead center. Any angle will serve with a degree 
of efficiency, as illustrated in Fig. 311 of Boehm's movement, where 
an extra link is added to destroy the dead point. Several links 
may be added. 

Another arrangement is shown in Fig. 312, where an extra crank 




Fig. 312. 

D, of radius equal that of A or B is added, together with the 
side extension AD and DB to the main link AB. The velocity- 
ratio is constant in the locomotive and in Figs. 311 and 312, the 



DEAD POINTS IN LINK-WORK. 275 

latter being obtained from Figs. 133 and 307 by cutting away the 
wheels and making the link parallel to the axis. 

In the steam engine two pitman links at about right angles 
are sometimes connected on the same crank pin, thus realizing con- 
ditions equivalent to the use of two cranks with parallel rods. 

Prof. Reuleaux has introduced a gab and pin at the points of 
the equivalent rolling curve where they are tangent to each other, 
when the linkage is at the dead centers. Thus in Fig. 306 a gab 
and pin is to be placed at C, where the hyperbolas are tangent, one 
on the link AD and the mate on the link BE. 

In case a longer link is fixed for the line of centers the gab and 
pin attached to it become fixed also, as in Fig. 313. 

Also, a gab and pin may be placed where the ellipses become 
tangent for the position of the line of centers, as shown in Fig. 314. 





Fig. 313. Pig. 314. 

Here also a gab and pin become fixed when a shorter link is fixed 
for the line of centers, as shown in Fig. 315. 

In Fig. 133 the gab and pin are in the form of a gear tooth and 
a space for the same, and represent the same case as Fig. 314, ex- 
cept that the elliptic wheel equivalents of the link- work are present 
and mounted with their respective links. 

In Fig. 305 a half gear tooth and half -space are made to serve 
for gab and pin to carry over the dead center. 

In Fig. 308 a gab and pin may be placed either at the vertices 
of the parabolas, or upon the links, as shown in Fig. 316, one set 
being sufficient. The dead center occurs when the swinging piece 
is vertical, since at that position the parts DE, FG, Fig. 308, may 
move up or down without control. One set of gabs and pins 
between A and D will prevent this. 



276 



PRINCIPLES OF MECHANISM. 



In Fig. 297 the curves show that a gab and pin may be placed 
at C as indicated by either system of curves. In Fig. 298 the pin, 
for instance, may be placed on AD extended downward from A to 
the curve at F, while the gab may be placed on the line of BE ex- 
tended to the curve at G. Thus the gab and pin are placed on the 





Fig. 315. 



Fig. 316. 



crank arms AD and BE, so that both are in motion. But a pair 
may also be placed on the longer members as at iif and 7", where the 
latter is fixed on AB. Thus two pairs of gabs and pins are avail- 
able in this linkage, either of which may be adopted, whichever 
member, as AB, AD, DE, or BE, is the fixed one, and in any case 
one, a gab or pin, may become fixed, as at /. 

Figs. 298, 299, and 300 show the possible arrangement of all the 
gabs and pins for this linkage. 

The linkage of Fig. 307 shows eight gabs and pins, any two pairs 
of which may be adopted in a particular case — sometimes all in 
motion, and sometimes two being fixed, as illustrated in Figs. 313 
to 315. 

In Fig. 316 four of the gabs and pins are at infinity; while of 
the four shown, one is fixed. 



Path of the Gab and Pin. 

In Fig. 317 is shown a link and crank connection recently 
adopted with success into a machine to transfer motion from one 
shaft to another parallel to it, the gabs and pins being employed. 
The connecting link is raised from its position a distance FG = DE 
to uncover the gabs at a and 1). 

The curves described by the centers of the pins a and b are 



DEAD POINTS IN LINK-WORK. 



2T 



drawn in, to show how the gabs must widen in amount of opening. 
The pins a and b are placed one-sided, for the reason that, if placed 
central, they would interfere with the gabs. But an examination 
of the path curves for a, b, and c shows that these curves are cusp- 




Fig. 317. 

shaped at the gab positions, and that the central one, c, does not 
differ materially from those at the sides at a and b. 

As to the advantage or disadvantage of placing the gabs some- 
what off of the line of centers, very little difference will be noticed. 

The position c is right for the rolling ellipses of Fig. 133, but 
here the eccentricity is so slight that interference occurs. 

The path curves for a, b, and c were traced by means of a paper 
templet on the drawing board, the templet being so cut that the 
edge fitted at the points F, a and D, and had a mark at each. This 
templet, placed in the various positions of F and D in the circles, 
and a noted for each, gave points through which the curves 
were traced. 

An example of a linkage nearly like that of Fig. 299 in applica- 
tion, is given in Fig. 318, which picks up the staple at the lower 




Fig. 318. 



end of the machine and delivers it at the upper end, thus represent- 
ing the handling of rods, screws, etc., in the manufacture of those 
articles. 



278 



PRINCIPLES OF MECHANISM. 



In Fig. 319 is given a model of link- work serving to prove that 
vibratory motion may be multiplied thereby. Thus the pointer is 
given four movements, to one of the first vibrating bar of the series. 




Fig. 319. 



CHAPTER XXV. 

LINK-WORK— (Continued). 

II. Axes Meeting. 

This is sometimes called conic link-work, or solid link-work, the 
principal essential consisting in bringing all the axial lines of shafts 



Fib. 320. 

and pins to a common point, 0, as in Fig. 320. Thus we may have 
equal cranks, as in the counterpart for parallel axes of Figs. 306 and 
307; or unequal cranks, as in Fig. 299. 

To realize the principles of equal cranks, it is only necessary to 
gives the cones AOD and BOB an equal slant. 

Any of the examples under axes parallel may be carried into 
conic link-work, even to the extent of continued trains. 

The Velocity-Ratio in Conic Link- Work. 

It will be advisable, if not necessary, to refer problems in conic 
link-work to spherical surfaces normal to all the axes, or which 
have O, Fig. 320, for the center of the sphere. 

In practice, avoiding spherical trigonometry, the velocity-ratio 

279 



280 



PRINCIPLES OF MECHANISM. 



may be most readily determined by preparing the proper spherical 

surface in wood or other material, 
and using it for the drawing board. 
Take Fig. 321 to represent this 
drawing board and drawing. 

A and B are the points where the 
axes A and B pierce this spherical 
surface; D and E where the crank- 
pin axes pierce the sphere; AB is 
the line of centers and DE the line 
of the link, both being parts of great 
circles, and constant in length on the 
sphere. 

Suppose A makes a slight turn, 
moving E to k, then D will move to 
I so far that the projections of Dl 
and Ek upon DE will equal each 
other and give Do equal to En, equal 
to pr, equal to st; since the spherical 
connecting link DE remains of constant length, s and p being 
points noted thereon where Ap and Bs are perpendicular to DE. 
Then the angle sBt measures the angular displacement of axis B, 
corresponding with that of pAr for A. But sB is a circle arc on 
the sphere, also Ap. Revolving s to u and p to v and transferring 
the arc Bu and Av to bd and ae in the great-circle section abO, we 
readily obtain the perpendiculars eg and lid, the inverse ratio of 
which is the velocity-ratio, because 




hd X angle sBt — st = pr — eg x angle pAr, 



giving 



velocity-ratio 



angle sBt 
angle pAr 



hd' 



Hence in practice, for any position of links as AEDB, draw the 
perpendiculars to the links Ap and Bs, which arcs transfer to ae 
and bd. Then the velocity-ratio equals the inverse ratio of eg 
and dh. 



The Rolling- Wheel Equivalent of Conic Link- Work. 

In Fig. 321 draw parallels ef and df to the axes Oa and Ob, and 
through f draw Oc. Then ac and be will be the spherical arc radii 



II. AXES MEETING. 281 

ior the spherical rolling-wheel equivalents of the link-work of Fig. 
321, for contact occurring when the links are in the positions 
shown. When the links are located for drawing the wheel, as for 
instance AE on the line of centers, their radii may be laid oil in 
place. Likewise for other radii, until a sufficient number of points 
iire determined for drawing in the wheel. 

The Dead Center and Gab and Pins in Conic Link-work. 

In Fig. 322 is given a photo-process copy of an example of conic 
link-work for equal cone slants, and where the angle between the 
axes A and B is equal to that be- 
tween the pins D and E when in 
one plane as in Fig. 320. 

Thus conditioned, the cranks 
will turn continuously in the same 
direction as in Fig. 312, or in op- 
posite directions, as in Fig. 313. 

The dead center accompanies 
conic link-work, and gabs and pins 
may be used here as well as in the 
oase of axes parallel. For Fig. 322, Fig. 322. 

the elliptic conic pitch lines of Prof. MacCord will serve to deter- 
mine the locations of the gabs and pins, those curves serving as the 
rolling-wheel equivalents of this linkage. 

Other linkages may have the rolling-curve equivalents deter- 
mined as in Fig. 321,- when the location of the gabs and pins is 
readily made. 

EXAMPLES OF PECULIAR MOVEMENTS. 

Example 1. — In Fig. 323 is what we might term a bent-shaft 
movement. A plunger is made to slide in the top head in a direc- 
tion parallel to the shaft. 

The joint work could be simplified by using a block and two 
pins, as in Fig. 324. If B is a square bar the center line of the 
pin D should strike the intersection 0. 

The velocity-ratio is the same as for the swash-plate movement 
shown in Willis, p. 172, where it is proved that the motion of the 
bar is the same as that of the crank and crosshead, with infinite 
pitman. 

The motion is still the same, if in place of the angular part of 
the shaft in Fig. 323 an enlarged bearing were used, like an ordi- 




282 



PRINCIPLES OF MECHANISM. 



nary eccentric and strap, and mounted on the straight shaft central, 
though at an angle like that of the swash plate. 

A movement like this has been used for working the valve in a 
steam engine exhibited at the Centennial of 1876. 




Example 2. — The Hooke's universal joint is often employed as 
a shaft angle-coupling, where the velocity-ratio is an important 
consideration. 

The joint is shown in Fig. 325, where A and B are the shafts 
to be connected, AD and FBE half- 
hoops between which is a cross with 
one branch at EF, parallel to the 
paper, and the other at D perpen- 
dicular to the paper. The branches 
of the cross are pivoted at E and F, 
and at the two points D, in AD. 

In the position shown the velocity 
of A is greatest, and the 




Fig. 325. 



velocity-ratio = 



DF 
aF' 



while at a 90-degree turn from the position shown the velocity of 
B is greatest when the 

velocity-ratio = -=q , 



II. AXES MEETING. 



28a 



because for the first position A acts in effect by an arm aF upon 
an arm DF from B, since aFis continually in the plane of the axis 
of A, and of the arm DF of the cross. 

If A revolves uniformly, the ratio of the fastest for B to the 
slowest, since db — aF, is 



Fastest for B 
Slowest for B 



(DFY 
\aF) 



These limiting speeds are in the same ratio as those for elliptic 
wheels of Fig. 133, where DF and AF are the distances from a 
focus to the remote and a nearer vertex respectively, though in the 
latter we have but one max. and one min. speed in a revolution, 
while in Fig. 325 we have two. But the law of variation of veloc- 
ity between extremes is not the same as for the elliptic wheels. 

In the case of a pair of two-lobed elliptic wheels, with the 
maximum and minimum diameters in the same ratio as DF to aF, 
the ratio of the limiting speeds is the same as for Fig. 325, and the 
number of changes in a revolution is the same, but the law of vari- 
ation of velocity still remains different. 

A pair of overhead shafts A and B connected by this joint, when 
the angle ADB differs much from 180 degrees, will be accompanied 
by too great a variation of velocity- 
ratio. 

In this case the use of two joints 
between A and B, arranged as in Fig. 
326, with the branches ^and HI 
of the crosses in the same plane 
and with the angle A OB = 2 GDO, 
will serve to transmit motion from 
A to B 9 with the velocity-ratio con- 
stant. 

For the single joint of Fig. 325 
the law of velocity-ratio is given by IG * 

the formula brought out in connection with Fig. 256, viz., 




ang. velocity of A 
ang. velocity of B 



sin 2 x . sin 2 BDJ 



cos BDJ 



in which x is the angle of movement of A from the positions shown. 

This equation may be constructed and solved graphically by the 

diagram, Fig. 327. Thus, draw BDJ with the same angle, BDJ y 



284 



PRINCIPLES OF MECHANISM. 



as in Fig. 325. Lay off DJ = 1 by some scale, and draw JL and 
L M perpendicular to DL and DM respectively. 




Then 



L R 
Fig. 327. 



JL = sin BDJ 



and 



JM = JL sin BDJ = sin 3 BDJ. 

Draw the semicircle JPM and lay off any angle x = J MP. 
Then 



and 



JM sin x = JP 
JP sin x = JQ = JM sin 3 x. 



Then 



DQ = DJ - JQ = 1 ~ JM sin 2 z = 1 - sin 8 a; . sin s BDJ. 
Draw the line PS, and we have 



i)£ = DQ sec £A7 = — ^— 
cos i?C/ 



Hence the 



velocity-ratio of A to B = Z)# =- 



sin 2 a . sin 9 £2)/ 



cos BDJ 

This supposes DJ — 1, but if it equals any other value, measure 
DJ and also DS by the same scale, and divide DS by DJ for the 

velocity-ratio. 



II. AXES MEETING. 285 

In a practical case draw BDJ, JL, LM and the semicircle JPM, 
as explained. This much is constant for a particular case of an 
angle BDJ. Then lay off all angles x the velocity-ratio is desired 
for, drawing the lines MN, MP, etc. Then project lines NR, PS, 
etc., perpendicular to DJ when we have the series of velocity-ratios 
DR, DS, etc. 

The velocity-ratio PL is a minimum and BD a maximum; the 
latter answering to the position shown, where x = 0. 

If A revolves uniformly, the fastest for B divided by the slowest 
gives 

F astest for B __ DB _ IDJV _ (DFy m . __. 
Slowest for B"~DL~ \DLj -\aFj 9 I* 1 **™) 

since from Fig. 327 

DL:DJ ::DJ: DB, 

or 

np S/' A DB (DJV 
DB = DL and DL = [dl)- 

Thus the results of Fig. 327 agree with that obtained from Fig. 
325. 

This example, the Hookers joint, is found in various forms of con- 
struction, the simplest being that for couplings for tumbling rods 
for transmitting power from the "horse-power" to the threshing- 
machine in agricultural districts. On each end of each rod is a 
forked casting, much like that in Fisr. 324, with bosses through 
which a pin may be placed at right angles to the rod. A block goes 
loosely between, with holes at right angles and just missing each 
other. Two pins or bolts are used at each joint, each passed through 
a fork and the block, and at right angles. Thus, between adjacent 
rods is a universal joint, so that the series of rods may lie upon the 
ground upon notched blocks, or on blocks and between stakes. 

. For a considerable angle between rods at a joint some end play 
will occur, when the holes for the bolts, as above, pass beside each 
other. 

Example 3.— The Almond, Eeuleaux, and other joints or coup- 
lings, are more compact than the Hooke's, so that they may very 
readily be enclosed in an oil-tight case, to facilitate lubrication. 

In Fig. 328 is the Almond coupling, serving to couple a pair of 



286 



PRINCIPLES OF MECHANISM. 



axes A and B at an angle, and with constant velocity-ratio equal 
to 1. 

At D is a fixed shaft, at the intersection of, and perpendicular 
to the plane of the axes A and B. On this shaft a sleeve slides 




Fig. 328. 



from which project two arms E, E, subtending the angle ADB, 
and upon the ends of each of which arms is a ball to work in a 
socket F in the swinging piece FH, the latter being in two pieces, 
held together as one by a bolt ab, and pivoted to J. 

When A revolves, each ball F is compelled to travel in a curve 
on the cylinder whose axis is D, one curve being identical with the 
other. Since both joined pieces FH are identical, it follows that 
the motion of B is a copy of that of A, and hence the velocity-ratio 
equals 1. 

This joint or coupling will connect shafts at any angle ADB, 
provided the angle between the arms EE is made the same. If the 
shafts A and B are in one straight line, the coupling will reverse 
the directions of motion. 

In Fig. 329 is a somewhat different joint or coupling, where the 
shafts A and B have fixed cranks with long crank pins, extending 
through the sockets or sleeves F, the latter having right-angled pin 
holes passing without meeting, through which are pins fixed in 
the arms E of the central piece. The latter slides on the fixed 
shaft or stud D, placed as before at right angles to A and B, 
and at their point of intersection. 



II. AXES MEETING. 



287 



It is readily seen that the parts D, EE, and FF compel the 
crank pin B to maintain the same relation to the shaft B as the 
crank pin of A does to the shaft A, so that the velocity-ratio is 
constant. 

The shafts may here also be connected at any angle from to 
180 degrees, or until interference of cranks occurs, the angle be- 
tween the branches EE being made the same as that between the 
shafts A and B. 

Besides this, the shaft B may be placed at any height above or 
below A by simply making D of suitable length and placing the 
arms E connecting with B the same amount above or below those 




Fig. 329. 

connecting with A as the one shaft is above or below the other. 
Thus by making D open, and extended into sliding and revolving 
gudgeons above and below, we meet the case of connecting by link- 
work axes which cross without meeting, and with a velocity-ratio 
unity. 

The Reuleaux coupling is shown in Fig. 330, where A and B 
are the shafts connected, C a head on their ends with holes at right 
angles to A or B for a joint pin, the latter passing the forked ends 
of EF at F, permitting the ends E to swing freely, which ends E are 
sleeve-like, receiving the round prongs of the head D, thus connect- 
ing EE through D. The ends of the prongs of D nearest F may 
have a nut and collar to prevent D from being thrown outward in 
revolving. 

This coupling or joint, so put up as to have the angle A FF equal 
the angle FFB, will have a velocity-ratio constant. 

This, as well as Figs. 327 and 328, should have firm bearings 



288 PRINCIPLES OF MECHANISM. 

for A and B, close to the movement, also in the former ones for D y 
an advisable arrangement being an iron framework especially for 
them. 

One important practical advantage of this latter over the former 




Pig, 330. 

two couplings is the fact that the mutual sliding of parts is simply 
from rotation upon pins, according to the true ideal in link-work, 
while in the former we have this combined with a disproportion- 
ately large amount of end sliding. 

At i^the branches of each part EF may be made unequal and 
one piece EF like another, so that the bosses of one piece EF will 
lie beside those of the other, uniting at F; and still admitting the 
four sockets E to all lie in one plane EEEE. 

In Fig. 329, as suggested by Prof. E. A. Hitchcock, we have a 
means of connecting the shafting in one shop room above or below 
another by link-work. 

In fact we may say any number of shafts parallel to one an- 
other in any one plane, by means cf one shaft D, Fig. 329, parallel 
to the plane, crossing all the shafts at any angle. 

This last statement corresponds with placing the shafts A and 
B, Fig. 329, with others in parallel in any plane, but at a distance 
from one another along the line of D, where the latter consists of 
a shaft free to slide endwise in bearings with a single-crank arm- 
piece E at each shaft A, B, etc., all keyed upon the shaft D in 
parallel, D being near the plane of the shafts and parallel to it. 
See Fig. 333. 



CHAPTER XXVI. 

LINK- W ORK— (Con tinued) . 

III. Axes Crossing Without Meeting. 

Ratchet and Click Movements. 

In Fig. 331 we have the typical case of projection of axes which 
are not parallel and not meeting, where A and £, A' and B' are the 
axes, Da and Eb the crank or lever arms, and ab the connecting 
link. At a and b some sort of universal joints are necessary, as, for 
instance, the ball-and-socket. 

According to Fig. 267 it would appear that the velocity-ratio 
may be obtained by giving the parts a small rotation, so that the. 




Fig. 331. 



end d of the common perpendicular clj makes the movement df> 
which, projected upon the connecting rod, gives effor its endloug 
component of displacement; also, that the end of the common per- 
pendicular gh makes the movement gi, which projected upon the- 
connecting rod gives hi for the endwise component of displacement, 
which components ef and hi must be equal, allowing permanence 
of length of connecting rod, so that the velocity-ratio will equal the 

289 



290 



PRINCIPLES OF MECHANISM. 



ratio of the angles djf to gki, each determined as the angle between 
two meridian planes containing the radii jd and jf, or kg and hi. 

To avoid the universal joint at a and ~b, Prof. Willis introduces 
a second intermediate piece between a and I, as at Tig. 332. 




Fig. 332. 

The shafts are shown as not parallel and not meeting, by the 
lines O'i', P' B f in elevation, and in plan by the lines OA and PB. 

At H is a crank for A, and Di? a connecting rod with the three 
axes meeting at 0. At 7 is a crank and GF& connecting-rod, with 
the three axes meeting at P. At J is an intermediate piece, with 
axes QP and QO to connect DE and GF. 

The parts thus connected may be moved about on their straight 
axes when remains a fixed point of the intersection of the axes 
AG, DO, and QO, while P remains a fixed point for the axes BP, 
OP, and QP. Also, Q is the point of intersection of the axes of 
J. In use these parts have long bearing surfaces without cramping 
or endlong sliding. 

This mechanism becomes somewhat complicated for connecting 
axes that do not meet. In many practical cases some sliding is not 
objected to, since the sliding facilitates lubrication, and by admit- 
ting sliding and rotary motion combined we often greatly simplify 
the mechanism, as in the following: 

Examples. 

Example 1. — In Fig. 333 we have the case of axes crossing ^ut 
not meeting, and yet have but one piece intermediate between the 
cranks upon the axes A and B, that piece serving as a link of in- 
finite length. 

The crank pins are parallel to the axes which support them. 



III. AXES CROSSING WITHOUT MEETING. 291 

As. the axes are both fixed in direction, it follows that the angle 




Fig. 333. 

between the pins is fixed, and that a block with long bearing holes 
at the same angle may be employed to connect the crank pins. 

Continuous motion is possible, except for the dead centers, and 
these may be passed by use of gabs and pins at the 
points determined in position, as by the equivalent 
non -circular gear pitch lines. 

Example 2. — In Fig. 334 is given an example 
which has been adopted in certain machines, with 
hundreds of them in successful operation for some 
years past. The axes A and B are at right angles 
and not meeting. 

The double sleeve D has holes at right angles 
fitting the pins closely. It turns out to be a 
simple, compact, and efficient movement, the piece 
D serving as a short link. 

Example 3. — In Fig. 335 is an example, in 
practical use in machines, of a case of axes A and 
B at right angles and not meeting, the piece D 
working on an eccentric A as driver, upon which 
it slides to accommodate it to the arc F about the 
fixed axis B, while D slides upon E to admit of 
the lateral movement of the eccentric. 

In practice, the longitudinal sliding of the 
knuckle piece D, combined with the rotary, is 
found an advantage, as favoring distribution of 
lubricant, and smoothness of surfaces in wear. 

Example 4. — Here link-work, much like that of Fig. 295, drives 




292 



PRINCIPLES OF MECHANISM. 



a pin F in a curve much like that of FE, while a lever pivoted at. 
B carries a pin G, between which and F is a knuckle piece D, 
working freely on both pins. 





Fig. 335. 



Fig. 336. 



As the pins are always at right angles, the piece D may have 
long bearing holes at right angles. 

If we take A of Fig. 295 as driver, and B, Fig. 336, as fulcrum 
of follower, these axes A and B cross without meeting. This is in 
successful operation in hundreds of tacking machines. 

Ratchet and Click Movements, Axes Variously Related. 

These movements are properly classed with link-work, some- 
times called intermittent link-work, because one end of the click or 
pawl is usually supported on a pin, while the other in action rests 
in a notch, which serves nearly as if pinned at that point. 

Example 1. — A Running Ratchet of simple form is shown in 
Fig. 337, such as was employed in the old-fashioned sawmills to 
feed along the carriage and log. A is a fixed pin about which AD 
swings, causing the end E of the click BE to move forward and 
back. As E moves forward the teeth of the wheel B are engaged 
and moved, while on the return E draws back over the teeth, the 
detent click F engaging a tooth and preventing the return of the 
wheel. 

Arranged as in the figure, D moves in the arc of a circle which, 
together with the movement of EB, causes ED to change position 



III. AXES CROSSING WITHOUT MEETING. 



293 



relative to EB, giving greater liability of B to slip out for one posi- 
tion than for others. Also, this change of position, or swinging of 
E in the notch, gives cause for wear. To reduce this, the pin D 
should be so placed as to swing in a normal to a line AB, or better, 




Fig. 337. 

a circle about B. An approximation to this, while using AD, is 
obtained by locating D to swing from H to 7, for which the position 
of DE with respect to EB is nearly constant for forward move- 
ment of E. 

In other cases D is mounted on a pin centered at B, or on the 
axis B, so that ED is fixed relative to EB when in action, as in 
the case of Fig. 339. 

Example 2. — A Running Ratchet for Varied Step Movement is 
shown in Eig. 338, as used in ther- 
mometer-plate graduating machines 
•over thirty years ago. At B is a ver- 
tical axis about which the slotted 
table GH swings. The slots ah, cd, 
etc., each have a stud that can be 
made fast at any point in the slot. 
These studs are slotted to receive a 
flexible ratchet strip DF, and are 
made fast in studs by thumb screws, 
as shown at J. A click, EE, as long- 
as the slots works the ratchet strip, 
and moves the table HG, notch by 
notch, between each of which move- Fig. 338. 

ments the graduating tool cuts a mark in the scale being gradu- 
ated, the slide supporting the scale, tool, etc., not being shown, 




294 



PRINCIPLES OF MECHANISM. 



the scale and its slide being moved in a straight line proportional 
to the angular displacement of HG. 

The ratchet strip DF is shown concentric with B, when it gives 
a uniform scale; but it may be set in any position, as dotted, on the 
table HG, when a correspondingly varied scale results— as finer at 
one end, or the other, or in the middle, or at both ends. 

This variability is required to meet the case of glass thermome- 
ter tubes which cannot be made of uniform calibre. 

Example 3. — A Reversible Running Ratchet is shown in Fig. 
339, where the click A is kept in engagement either way by a 
spring action at C. A central notch for C holds the click out of 
engagement when desired. 

A variety of forms of A are in use, especially in feed motions 
for machinists' tools. 

Example 4. — A Reversible Varied Rate Running Ratchet is 
shown in Fig. 340, where G is continually reciprocating about B, 
carrying the pawls or clicks E and F forth and back on the click 





Fig. 340, 



guard D. As H moves D to the one side or the other a click en- 
gages one or several teeth, and moves B little or much, according 
to extent of movement of D. 

When D moves to the left the wheel B is turned right-handed 
by the click E, or left-handed by the click F if D is thrown to the 
right. The more D is displaced, the more rapidly will B be moved 
by the click, and when central, B is stationary. 

This movement is found in Snow's "Waterwheel Governor. 



RATCHET MOVEMENTS. 



295 



The click guard D has had frequent application with good 
results. 

Example 5. — A Continuous Running Ratchet is shown in Fig. 
341, where the wheel is moved the same way for each movement of 
the handle J one way or the other. Thus the holding or detaining 
click of Fig. 337 is here made a working click. The clicks may 
push or pull in action, the former being usually preferred for con- 
siderations of strength and shown above in full lines, while the 
latter are dotted in. The pins D and E are sometimes placed in a 
line perpendicular to AB, but the figure gives that relation which 
causes least turning of the ends of the clicks in the teeth, and 
consequently least wear. 

The teeth may be internal as a possibility, but are usually ex- 
ternal. 

Forms of Teeth. 

At D, Fig. 342, the normal d to the working surface a of a tooth 





Fig. 341. Fig. 342. 

may go inside of D, when the click will be secure. But if it goes out- 
side of D, as for the case that b were the working face of the tooth, 
the click will be insecure, or very likely to fly out of engagement. 

Again, if Dd is excessive, the wheel will turn back appreciably 
before coming to bearing against the click rafter it drops in. 

The click may have a shape for engagement with a common 
gear tooth, as at F. 

At G several clicks may be placed on the same pin, and of dif- 



296 



PRINCIPLES OF MECHANISM. 



ferent lengths, one engaging after another and serving in effect to 
divide the tooth pitch into parts answering to a finer toothed 
ratchet wheel. This may be called a differential Ratchet, 

At E is a stationary ratchet click serving to hold the wheel 
stationary against moving either way. 

Friction Ratchets. 

In Fig. 343, E may represent a round rod and D a washer easily 
sliding over E, except when cramped, as by the pull F : when the 
greater the pull the greater the binding or ; ^rip 
upon E 9 so as to hold effectually^ This form of 
ratchet has been used in the Brush electric 

aD light lamp. 

The part E may be a square rod or a flange 
on a wide piece, D fitting properly. 

This form of ratchet always works satisfac- 
torily when new, but in practice very soon be- 
comes untrustworthy, and is usually abandoned, 
Fig. 343. especially under heavy service. 

Shoe pieces will extend the wearing surfaces, as in Fig. 344, and 
they have been used with fluted seats, as shown in plan. 

These work admirably for a much longer time than the more 
limited bearing surfaces of Figs. 342 or 343, but finally become 






Fig. 344. 



worn in places where most used, and fail to work as expected. 

Example 1.— A Continuous Friction Ratchet is shown in Fig. 
345, which was once used on a rock-drilling machine for feeding 
the working parts along as the drilling progressed. At D is a 



RATCHET MOVEMENTS. 



297 



flange fast to the stationary framework, while J is a part oi the 
sliding carriage, G and F being the grip clicks, one holding in the 




Fig. 345, 

opposite sense of the other, H is a clamp nnder slight friction grip, 
that can be moved towards G as the drilling advances and a tappet 
comes to press upon it. That moves G, and in turn F moves by the 
spring I. F prevents J from moving up, and G opposes its down- 
ward movement until H acts upon it. 

This apparatus operated most admirably for a few days, but soon 
had to be abandoned, as most ratchets of this kind. 

Example 2. — Wire Feed Ratchets, as in Fig. 346, with teeth to 
grip the wire, the lower ones reciprocating to- 
gether while the upper ones are stationary, 
always operate with satisfaction for a few 
hours, for feeding wire, when the teeth become 
worn and slip, making the contrivance a failure. 
Example 3. — Running Face Ratchets have 
C o \ ? s ) *^ e teetn u P on tne sides of the wheel instead of 
S-^^x v_^~^ edge, as shown in Fig. 347. 
In this example, recently 
adopted with success in 
practice, the click E is a 
full circle, as well as the 
having the same number of 




Fm, 346. 




wheel Z>, both 
teeth, viz., 100. 

Ratchet Gearing is made in great variety — 
too much so to attempt full description here. Fig. 347. 

An elaborate enumeration and illustration will be found in Reu- 
leaux's Constructor, translated by Suplee. 



PART V. 



CHAPTER XXVII. 



KEDUPLICATION. 



This term is applied to such movements as consist of circular 
< pulleys or sheaves, and flexible connectors 
passing over them, usually made fast at one 
end, mostly ropes, employed to multiply the 
pull, or the motion. Blocks and tackles of 
ships are included. 

In Fig. 348 is a simple example where the 
rope is made fast at G, passes down around 
the pulley F, returns and passes over the 
pulley E, and on to some point D, where a 
pull may be exerted to raise a weight at F, as 
in the case of hay unload ers, stackers, etc. 

When the ropes from F up are parallel, the 
velocity-ratio is constant; but when the rope 
diverges or converges upward, the velocity-ratio is variable, that is, 
the velocity of F as compared with that of D. 

In this case D must move twice as far as F, and the pull exerted 
at F will be twice as great as at D. 

Again, by making F the driver, D will be moved twice as far as 
F, and the pull exerted at D will be one half that at F, so that the 




Fig. 348. 



velocity -ratio = 2, or 1/2. 



In Fig. 349, i^is raised faster than in Fig. 348. To find the 
velocitv-ratio, draw abed with lines parallel to the principal figure. 



REDUPLICATION. 



290 



Then if D is pulled a distance cbd, F is raised a height db, be- 
cause ac is perpendicular to GF, and represents the movement of 




a point of 67i^near F, when c5d is removed. Hence, for Fig. 349, 

velocity of D _ cbd _ 2FH 

velocity of F ~ ab ~ EF ' 

Applying this to the case of Fig. 348, we will find EF = FH, and 
the velocity-ratio reduces to 2, as before 
stated. One make of a hay-unloader is 
as shown in Fig. 350, where the velocity- 
ratio is 3 to 1 and constant, when the 
ropes are parallel as shown. 

In the Case of Ships' Tackle, where 
several sheaves are placed in a block with 
the ropes parallel, the velocity-ratio can 
readily be made out on the same principle 
as in the above figures. The ropes are 
usually nearly parallel in blocks and 
tackle, so that the velocity-ratio is nearly 
constant. 

In Elevators operated by hydraulic 
power, the lifting rope passes from the cage up over a pulley at the 
top of the shaft or hoisting way, and down to a series of sheaves, 
and finally made fast to a stationary hitch at the lower end. 

If, for instance, D, Fig. 351, represents the cage, G a set of 
stationary sheaves, F a set of sheaves attached to the hydraulic- 
power rod at F, then when F is drawn back D will be elevated. 

Suppose the rope passes from the hitch near G to and around 
F, and then over to G, up to E, down to D, thus placing two ropes 
between the pulleys F and G. Then when F is drawn back 10 feet 
D will be elevated 20 feet, the multiplying being as many times as 
the ropes between G and F. The raising of D will be 80 feet for 




Fig. 350. 



300 



PRINCIPLES OF MECHANISM. 



8 ropes between F and G, and for a draft movement of 10 feet for 
F. The velocity-ratio will be 8 at the same time.- 

I -A 



¥) 



G 



-0ZH 



Fig. 351. 

The Weston Differential Block, shown in Fig. 352, is a simple 
device for one so powerful. 

An endless chain passes around F, then up aud around E, then 
down in a loose " fall " D, then around G, slightly larger than E, 




Fie. 352. 




and down to F again ; G and E being in one piece. 

A chain is used instead of a rope, to serve as a hitch at EG, 
which are sprocketed to hold to the chain. Thus there is a hitch, 
in some measure equivalent to the rope hitch in the preceding 
cases, so that the device may be classed here. 



REDUPLICATION. 301 

Fig. 353 shows that when D is pulled a distance HG, L is 
raised by the amount GH } while K is lowered at the same time 
by the amount J I, and F is raised 1/2 {GH — IJ), so that the 
velocity-ratio, as between D and F } is 

velocity of D _ 2HG 
velocity oO'~ HG - IJ T ' 

if i? and r are the radii of the two sizes of sheaves, EG and EL 
If R = 5" and r = 4", then the velocity- ratio equals 10. 



INDEX. 



PAGE 

Addendum, dedendum, or root line 93, 107, 148, 161, 175, 178 

Alternate motions, circular, teeth for, limited 190 

, unlimited 191 

, limited and unlimited mangle-wheels 83-86 

, pitch lines for. ... » 83 

.teethfor.. 123 

Annular wheels, epicycloidal teeth for 141 

, interference 141 

, involute teeth 144 

, interference 144 

Approximate gear teeth 155-162 

Belt gearing, any position of pulleys 247 

, barrel and fusee for chronometers 240 

, chain and sprocket 256 

, cone pulleys for lathes, etc 247 

, draw-bridge equalizer 239 

, for treadle 243 

, gas-meter prover equalizer 238 

, law of motion given to find the wheels 237 

, non-circular, for rifling machine 241 

, pulley with high center. . , 246 

, quarter twist, guide pulley 246 

, retaining belt on pulley 245 

, rope transmission 251-254 

, spinning- mule snail, etc 241 

, the belt and circular pulley 244 

, velocity-ratio, circular and non-circular 236 

Bevel gearing, circular, epicycloidal, involute 175 

, circular, pitch lines for 8 

, intermittent motion 82 

, non-circular pitch lines 62-74 

. teeth for non-circular wheels 113 

Cams, in general , „ t . 192 

, by co-ordinates 192 

intersection 193-195 

303 



304 INDEX. 

PAGE 

Cains, conical 198 

, roller for 216, 217 

, cylindric, straight path 196- 

, curved " .. 197 

, diameter constant 206, 213 

, bread th " , 207,209 

, easement for 208, 204 

, headdle cam 202 

, inverse, velocity-ratio , 21& 

,• law of motion defined . . 199-202 

, return of follower by gravity 211 

by spring, cam groove, positive 212 

, roller, and its pin for 214, 217 

, for spherical cam 217 

, best form 216 

, at salient points : 215 

, spherical 198 

, tarrying points 202 

, thick and irregular follower extremity 215 

, uniform motion for. 202 

, velocity-ratio 194 

, with flat-footed follower 199, 205, 208 

several followers 208 

Circular rolling wheels, pitch lines: 7 

, teeth for ... 128 

Chain and sprocket gearing 256 

Clearing curve and filleting 149 

, possible 149 

Conical cams 198 

link-work 279 

pulleys in belt gearing 247 

roller for cams 217 

wheels for gears, circular '. 8 

Conjugate gear teeth for circular wheels , 145-147 

non-circular wheels. . . 102 

, Sang's theory for ; 168 

Couplings, Almond > 286* 

, axes not meeting, for .' 287 

, Hooke's 283 

, Oldham's 220-224 

, Reuleaux's 288 

Cutters for gear teeth, epicycloidal and involute. '. 167 

Cycle '. 3 

Dead-points, in link- work 274 

conic link- work 281 

Directional relation 4 

, constant, variable 4, 7 



INDEX. 305 

PAGE 

Elliptic wheels, pitch lines for. . . * 30 

, interchangeable multilobes 33-35 

Epicycloidal curves, peculiar properties 129 

engine, to form gear- tooth curves. . 164 

gear teeth, annular wheels 140 

, flanks radial 130 

concave 132 

convex 132, 140 

, for inside pin gearing 136 

, interchangeable sets 133, 140 

, interference of annular gears 141 

, least crowding and friction j 37 

, line of action 151 

, pin annular wheels 136, 1H7 

, and pin teeth 134 

, rack and pinion 137-139 

, two-tooth pinion 136, 137 

tooth cutters 167 

Escapements, anchor, for clocks, dead-beat 226 

, chronometer, for watches and chronometers 234 

, cylinder, for watches 231 

.duplex, " " 233 

, gravity, for clocks, dead-beat 229> 

, lever, for watches 232 

, pin-wheel, for clocks, dead-beat 228 

, recoil, " " 227 

, power escapements 225 

Friction wheels, pitch lines for. 7 

in cams 21 3 

, relieved by roller 214 

ratchets 296 

Grant's gear-cutting engine 168, 170 

Hindley's corset-sbaped worm 188 

Hooke's universal joint or coupling 283 

Hyperbolic wheels, pitch lines 36 

Hyperboloids, circular rolling 10 

Intermittent motions, circular, locking arcs IS 

, easements 13 

, spurs and segments 13 

, bevel and skew bevel 189 

, non-circular, plane and bevel 81, 82 

, rolling spurs for 120 

, solid easement segments 122-125 

, teeth, spurs, and segments 116-127 

Involute bevel gear teeth 175 



306 INDEX. 

PAGE 

Involute gear teeth, cutters for 167 

, described with log-spiral 142 

on drawing-board 155 

, line of action for 151 

Line of action of pressures between teeth „ . . 151 

in belt gearing 237 

centers, contact 5 

Link- work, axes crossing without meeting 289 

, or skew- bevel link -work, velocity-ratio 289 

with sliding blocks 291,292 

of Willis 290 

, axes parallel 259 

, Corliss valve motion 260 

1 /nailing machine motion 261 

, needle bar motion 260 

, conic, or solid, velocity- ratio 279 

, Almond's . . 286 

, examples 282-288 

, gab and pin, and dead center 281 

, for axes not meeting 287 

, Hooke's joint, and velocity for 282, 283 

, rolling wheel equivalent for 280 

, Reuleaux's 288 

, dead points in 274 

, gabs and pins to pass ... 264, 275 

, path for 276 

, path and velocity of various points 261 

, peculiar features 259 

, rolling curve, equivalent for 264 

examples 264-273 

as ellipses 271 

for crank and pitman 267 

as hyperbolas 270 

as link and drag link 268 

for parabolas 272 

Sylvester's kite 268 

unsymmet rical, example 273 

, sliding blocks and links for 263, 291 

, velocity-ratio for 258 

Machine, and parts of 1 

-made gear teeth 164 

gear-tooth cutters * . 166, 167 

Mangle wheel and rack, pitch lines for 84-86 

, teeth for 127 

Mechanism, elementary combinations of 1 

, primary and secondary trains of 1 



INDEX. 307 

PAGE 

Mechanism, train or trains of. 1 

, table of elementary combinations of 2 

Motion, uniform, angular 3 

Names and terms in mechanism 3 

gearing 93 

Non-circular wheels, plane , , 19 

, equal log-spiral, segmental 20-24 

, blocking tendency 107 

, in general 44 

, for intermittent motions 81 

, internal or annular Ill 

, involute, teeth for 101 

, laws of motion given, to find the wheels. . . 48-53 

, limit of eccentricity 106 

, log-spiral wheels, complete 24-29 

, one wheel given, to find its mate 44-47 

, rolling, in extreme eccentricity 109 

, five special forms 20 

, solutions of practical problems 54-61 

, bevel, drawn direct on normal sphere 73 

, example 72 

, in general, solution 69 

, five special forms. 62-66 

, teeth for 112 

, teeth for skew-bevel 114 

Normal sphere, drawing wheels directly on 62-65, 71, 76 

Odontograph, Grant's 161 

, templet 157 

, Willis' ' 158 

Olivier spiraloid teeth for skew bevels 182 

, contact between . 183 

, flat faces away from gorge 185 

, interchangeability of 182 

, interference of 183 

, practical at gorge 187 

, result of example 184 

Parabolic wheels, equal and similar. 35, 272 

, equivalent link-work 272 

, transformed 41 

, with gabs and pins 276 

Path of contact in gearing 149-150 

, limited 151 

Period 3 

Pin and slit movement, inverse cam 219, 220 

Pitch lines 7 



308 INDEX, 



PAGE 



Pitch lines, rolling in non-circular wheels 109, 110 

, circumf erential 94, 148 

, diametral 94, 148 

Point of contact 5, 7 

, between the axes 7 

.outside " " 8 

Practical considerations in circular gearing 148-163 

Rack and pinion, epicycloidal teeth 139 

, involute " 144 

Radius, rod to carry templet odontograph 157 

tooth templet 154 

Ratchet and click, movements 292 

, form of teeth and click 295 

, running . 292 

, continuous 295 

, face ratchets 297 

, friction ratchets 296 

, wire feed 297 

, reversible 294 

, varied rate 294 

, for varied steps 293 

Return of follower in cams, by gravity 211 

, by spring, by second cam, by groove 212 

Reduplication, velocity- ratio , 298 

, elevator pulley and rope 300 

, parallel ropes 298, 299 

, ropes not parallel 298 

, Weston's differential block 300 

Revolution, a complete turn 4 

Rolling contact 5 

curve, equivalent for link work 264 

hyperboloids 10, 75 

Roller, in cam movements 214 

, proper form of 216 

, pin for 217 

Rope transmission, short and long stretch 253* 

, for haulage lines 254 

Rotation, a partial turn 4 

Sang's theory of conjugating gear-teeth 168 

Skew-bevel wheels, circular pitch surfaces 10-13 

, error in early statement of 11 

, non-circular to given law 76 

, pin teeth , 188 

link-work 289 

Sliding contact, in general, velocity-ratio 87-90 

and rolling contact, in one model 90 



IKDEX. 309 

PAGE 

Speeds, geometrical series of speeds in cone pulleys 247 

Swasey's gear-cutting engine 168 

Teeth of gear wheels, conjugate 102, 145 

, blocking tendency 107, 152 

, for bevel and skew-bevel non-circular 112-115 

intermittent and alternate motions 116-127 

involute non-circular 101 

, individually constructed 101 

, generation of tooth curves for , 92-105 

templets for 95-101 

, limited inclination, hooking 103-105 

, names and rules 93 

- , short, in eccentric wheels. 107, 152 

, trachoidal 99 

Templets, for generating gear teeth, form and size 95-100 

, motion templets 127 

, odontograph 157 

, path templets 193 

, tooth " 154 

Tooth profile, epicycloidal, for circular wheels 128, 152 

, involute, " " " 143,155 

, for non-circular wheels 98 

Transformed wheels, and examples 38 

, unilobed 39-42 

, bevel, several examples , 66-68 

Velocity, angular 3 

, velocity-ratio. 3 

, in rolling contact 5 

•ratio in belt-gearing 237 

link-work 258, 279, 289 

reduplication 298 

sliding contact 87 



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Cromwell's Toothed Gearing 12mo, 1 50 

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Du Bois's Mechanics. Vol. L, Kinematics 8vo, 3 50 

Vol. II., Statics 8vo, 4 00 

Vol. III., Kinetics 8vo, 3 50 

Dredge's Trans. Exhibits Building, World Exposition, 

4to, half morocco, 15 00 

blather's Dynamometers 12mo, 2 00 

" Rope Driving 12mo, 2 00 

Richards's Compressed Air 12mo, 1 50 

Smith's Press-working of Metals 8vo, 3 00 

Holly's Saw Filing 18mo, 75 

Fitzgerald's Boston Machinist 18mo, 1 00 

Baldwin's Steam Heating for Buildings 12mo, 2 50 

.Metcalfe's Cost of Manufactures 8vo, 5 00 

Benjamin's Wrinkles and Recipes 12mo, 2 00 

Dingey's Machinery Pattern Making 12mo, 2 00 

METALLURGY. 

Iron— Gold— Silver — Alloys, Etc. 

.Egleston's Metallurgy of Silver 8vo, 7 50 

Gold and Mercury 8vo, 7 50 

" Weights and Measures, Tables 18mo, 75 

" Catalogue of Minerals 8vo, 2 50 

CDriscoll's Treatment of Gold Ores 8vo, 2 00 

* Kerl's Metallurgy — Copper and Iron 8vo, 15 00 

* << » Steel, Fuel, etc 8vo, 15 00 

12 



Thurston's Iron and Steel 8vo, $3 50 

" Alloys 8vo, 2 50 

Troilius's Chemistry of Iron 8vo, 2 00 

Kunhardt's Ore Dressing in Europe 8vo, 1 50 

Weyrauch's Strength of Iron and Steel. (Du Bois.) 8vo, 1 50 

Beardslee and Kent's Strength of Wrought Iron 8vo, 1 50 

Compton's First Lessons in Metal "Working 12mo, 1 50 

West's American Foundry Practice 12mo, 2 50 

" Moulder's Text-book 12mo, 2 50 



MINERALOGY AND MINING. 

Mine Accidents— Ventilation— Ore Dressing, Etc. 

Dana's Descriptive Mineralogy. (E. S.) 8vo, half morocco, 

" Mineralogy and Petrography. (J. D.) 12mo, 

" Text-book of Mineralogy. (E. S.) 8vo, 

" Minerals and How to Study Them. (E. S.) 12mo, 

" American Localities of Minerals 8vo, 

Brush and Dana's Determinative Mineralogy 8vo, 

Rosenbusch's Microscopical Physiography of Minerals and 

Rocks. (Iddings.) 8vo, 

Hussak's Rock- forming Minerals. (Smith.) 8vo, 

Williams's Lithology 8vo, 

Chester's Catalogue of Minerals 8vo, 

" Dictionary of the Names of Minerals 8vo, 

Egleston's Catalogue of Minerals and Synonyms 8vo, 

Goodyear's Coal Mines of the Western Coast 12mo, 

Kunhardt's Ore Dressing in Europe 8vo, 

Sawyer's Accidents in Mines 8vo, 

Wilson's Mine Ventilation 16mo, 

Boyd's Resources of South Western Virginia 8vo, 

" Map of South Western Virginja Pocket-book form, 

Stockbridge's Rocks and Soils 8vo, 

Eissler's Explosives— Nitroglycerine and Dynamite 8vo, 

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12 50 


2 00 


3 50 


1 50 


1 00 


3 50 


5 00 


2 00 


3 00 


1 25 


3 00 


2 50 


2 50 


1 50 


7 00 


1 25 


3 00 


2 00 


2 50 


4 00 



*Drinker's Tunnelling, Explosives, Compounds, and Rock Drills. 

l4to, half morocco, $25 00 

Beard's Ventilation of Mines 12mo, 2 50 

Ihlseng's Manual of Mining 8vo, 4 00 

STEAM AND ELECTRICAL ENGINES, BOILERS, Etc. 

Stationary— Marine— Locomotive— Gas Engines, Etc. 

Weisbach's Steam Engine. (Du Bois.) , 8vo, 5 00 

Thurston's Engine and Boiler Trials 8vo, 5 00 

Philosophy of the Steam Engine 12mo, 75 

Stationary Steam Engines 12mo, 1 50 

Boiler Explosion 12mo, 150 

Steam-boiler Construction and Operation 8vo, 

Reflection on the Motive Power of Heat. (Carnot.) 

12mo, 2 00 
Thurston's Manual of the Steam Engine. Part I., Structure 

and Theory 8vo, 7 50 

Thurston's Manual of the Steam Engine. Part II., Design, 

Construction, and Operation 8vo, 7 50 

2 parts, 12 00 

Rontgen's Thermodynamics. (Du Bois. ) 8vo, 5 00 

Peabody's Thermodynamics of the Steam Engine 8vo, 5 00 

" Valve Gears for the Steam-Engine 8vo, 2 50 

Tables of Saturated Steam 8vo, 1 00 

Wood's Thermodynamics, Heat Motors, etc 8vo, 4 00 

Pupin and Osterberg's Thermodynamics 12mo, 1 25 

Kneass's Practice and Theory of the Injector .8vo, 1 50 

Reagan's Steam and Electrical Locomotives 12mo, 2 00 

Meyer's Modern Locomotive Construction 4to, 10 00 

Whitham's Steam-engine Design 8vo, 6 00 

" Constructive Steam Engineering 8vo, 10 00 

Hemenway's Indicator Practice 12mo, 2 00 

Pray's Twenty Years with the Indicator Royal 8vo, 2 50 

Spangler's Valve Gears f 8vo, 2 50 

* Maw's Marine Engines Folio, half morocco, 18 00 

Trowbridge's Stationary Steam Engines 4to, boards, 2 50 

14 



Tord's Boiler Making for Boiler Makers. 18mo, $1 00 

^Wilson's Steam Boilers. (Flather.) 12rno, 2 50 

Baldwin's Steam Heating for Buildings 12mo, 2 50 

Hoadley's Warm-blast Furnace 8vo, 1 50 

Sinclair's Locomotive Running 12mo, 2 00 

^Clerk's Gas Engine t 12mo, 4 00 

TABLES, WEIGHTS, AND MEASURES. 

For Engineers, Mechanics, Actuaries— Metric Tables, Etc. 

Crandall's Railway and Earthwork Tables 8vo, 1 50 

Johnson's Stadia and Earthwork Tables 8vo, 1 25 

Bixby's Graphical Computing Tables Sheet, 25 

Compton's Logarithms 12mo, 1 50 

Ludlow's Logarithmic and Other Tables. (Bass.) 12mo, 2 00 

Thurston's Conversion Tables 8vo, 1 00 

^Egleston's Weights and Measures 18mo, 75 

Totten's Metrology 8vo, 2 50 

Wisher's Table of Cubic Yards Cardboard, 25 

-Hudson's Excavation Tables. Yol. II 8vo, 1 00 

VENTILATION. 

Steam Heating — House Inspection — Mine Yentilation. 
t 

-Beard's Yentilation of Mines 12mo, 2 50 

Baldwin's Steam Heating 12mo, 2 50 

Heid's Yentilation of American Dwellings 12mo, 1 50 

"Hott's The Air We Breathe, and Yentilation 16mo, 1 00 

Oerhard's Sanitary House Inspection Square 16mo, 1 00 

"Wilson's Mine Yentilation 16mo, 1 25 

•Carpenter's Heating and Yentilatingof Buildings 8vo, 3 00 

niSCELLANEOUS PUBLICATIONS,, 

JUcott's Gems, Sentiment, Language Gilt edges, 5 00 

3ailey's The New Tale of a Tub , 8vo, 75 

Ballard's Solution of the Pyramid Problem 8vo, 1 50 

•Barnard's The Metrological System of the Great Pyramid. .8vo, 1 50 

15 



* Wiley's Yosemite, Alaska, and Yellowstone 4to, $3 00' 

Emmon's Geological Guide-book of the Rocky Mountains. .8vo, 1 50' 

Ferrel's Treatise on the Winds 8vo, 4 00< 

Perkins's Cornell University Oblong 4to, 1 50 

Ricketts's History of Rensselaer Polytechnic Institute 8vo, 3 00- 

Mott's The Fallacy of the Present Theory of Sound. .Sq. 16mo, 1 00 
Rotherham's The New Testament Critically Emphathized. 

12mo, 1 50 

Totten's An Important Question in Metrology 8vo, 2 50 

Whitehouse's Lake Moeris Paper, 

HEBREW AND CHALDEE TEXT=BOOKS. 

For Schools and Theological Seminaries. 

Gesenius's Hebrew and Chaldee Lexicon to Old Testament. 

(Tregelles.) Small 4to, half morocco, 5 00 

Green's Grammar of the Hebrew Language (New Edition). 8 vo, 3 00 

'• Elementary Hebrew Grammar. 12mo, 1 2.S 

" Hebrew Chrestomathy • • • • 8vo, 2 00 

Letteris's Hebrew Bible (Massoretic Notes in English). 

8vo, arabesque, 2 25 
Luzzato's Grammar of the Biblical Chaldaic Language and the 

Talmud Babli Idioms 12mo,^ 1 50 

MEDICAL. 

Bull's Maternal Management in Health and Disease 12mo, 1 00 

Mott's Composition, Digestibility, and Nutritive Value of Food. 

Large mounted chart, 1 25 

Steel's Treatise on the Diseases of the Ox. . . 8vo, 6 00 

" Treatise on the Diseases of the Dog 8vo, 3 50 

Worcester's Small Hospitals— Establishment and Maintenance, 
including Atkinson's Suggestions for Hospital Archi- 
tecture 12mo, 1 25 

Hamvntnsten £ Physiological Chemistry. (Mandel.) .8vo, 4 00 



16 



